How do forces 'know' they need to move when a system is in motion? I am curious as to how forces move when a system is in motion. This was never fully explained in my physics classes at university. Let me explain:
I understand the Newtonian (classical) physics that there are equal and opposite forces in play.  So when I am standing the force I am exerting on the ground due to gravity is balanced by an upward force from the ground.  However, when I lift one foot (say the left foot) the force from my body is now transferred through the right foot. However, where did the upwards force that was under my left foot go?
I assume the upward force 'moved' to balance the increased force exerted by my right foot. I can understand that it general, except for one point. How did the upward force 'know' that it needed to move - and, secondly, where it needed to move to?
This same question can be applied to many dynamic situations of motion, such in a moving vehicle.  (I can think of many other examples as well).
I had one physicist trying to explain it to me but, I admit, I lost his explanation when he went down the quantum mechanics rabbit hole. Is there a classical explanation as to how forces know when and where to move when a system is in motion?
 A: The force does not move. There is a difference between a force "moving", and a force changing. Both forces changed, the one in the foot you lifted, and the one on the foot on the ground. As you move you feet up, the distribution of matter changes, and with it the contact forces between you and the floor. This happens instantaneously in Newtonian mechanics, no quantum mechanics need to be invoked. Just doing a force diagram of the new situation will determine what are the strength of all forces in a given configuration of your body. However, I am not sure this helped to clarify your confusion at all.
A: 
So when I am standing the force I am exerting on the ground due to gravity is balanced by an upward force from the ground. However, when I lift one foot (say the left foot) the force from my body is now transferred through the right foot. However, where did the upwards force that was under my left foot go?

If you feel balanced by the upwards force on both feet, and now simply try lifting e.g. your left foot, you'll start tilting to the left side. Continue doing this, and you'll fall down.
The actual process of lifting one foot (for concreteness, left) so as to avoid falling down instead consists of the following (simplified) sequence:

*

*Do a jumping motion by the left foot or leg, which makes your body tilt a bit rightwards.

*Do the lifting motion by the left leg.

After this sequence your weight has been transferred to your right leg, after which you lifted the left one, which was no longer under load.
As you can see, it's the initial actions that you do to avoid falling down that "inform" the ground of the change.
A: Forces are not considered to move in the sense you describe. Basically, a force is not like energy. If you have energy in one location then it can be localized and tracked as it moves to another location. We call such quantities (locally) conserved. Force is not conserved. It can just appear or disappear as needed to satisfy the laws of physics.
This can be understood purely classically with no need to invoke quantum mechanics. Forces are simply governed by various force laws. In the scenario you mentioned the relevant laws are Newton’s laws, the law of gravity, and Hooke’s law.
Newton’s laws describe how forces act in general. In your scenario they require that the force that the force of the floor pushing up on your foot must be equal and opposite to the force of your foot pushing down on the floor. They also say that your acceleration is the sum of the gravitational force pulling down and the contact force pushing up (divided by your mass).
It sounded like you were already aware of those. The force law that you may not be aware of is Hooke’s law. He said that the force required to deform an elastic object is proportional to the distance you deformed it. When you stand on the floor it pushes you up, but you push it down. This leads to small deformations in both the floor and your shoe.
When you lift one foot up, the force does not move from one foot to the other. Instead, you simply deform the floor a little more under your other foot and by Hooke’s law that increases the force pushing up there.
If you are worried about how the floor knew to deform more, consider what would have happened if it didn’t. Then as the first foot was lifted up, the net force would no longer balance, so your other foot would begin to accelerate into the floor. Since the foot and the floor cannot occupy the same space this would push the floor down, thus  deforming it. So regardless, it deforms.
A: Classical physics is an approximation to reality, but it works just fine for everyday use. For the most part, the world is made up of three kinds of point particles: electrons, protons, and neutrons.
All of these particles attract each other with a weak gravitational force.
There are also much stronger electrical forces. Electrons repel electrons. Protons repel protons. Electrons and protons attract each other. When electrons and protons move, there are also magnetic forces. Neutrons are not affected by electric forces, and have a very weak response to magnetic forces.
There are also two more forces: The strong and weak nuclear forces. For the most part these are ignored. They make protons and neutrons attract each other when extremely close together and stick to each other in atomic nuclei. Other than this, these forces are mostly ignored in classical physics.
There is also light. Light is an electromagnetic wave, an electric/magnetic field that varies in time and space.
For the most part, that is the world of classical physics. There is more, but when people talk about it, they usually get into better approximations of reality like relativity and quantum mechanics.
Classical physics doesn't explain why there are forces. They just are. It explains in detail how forces affect particles, and how particles and their motion affect the forces.
One important thing about these forces is they occur between pairs of particles. If particle A exerts a force on Particle B, then Particle B always exerts an equal and opposite force on particle A.
Explanations are simpler when speeds are much much slower than light, and this suffices for your question. Forces are simple attractions and repulsions. All the particles in you are gravitationally attracted to all the particles in the Earth. Since there are a lot of particles in the Earth, this weak gravitational for adds up.
There are more complex forces between atoms, where there is both attraction and repulsion at the same time. It takes quantum mechanics to explain it correctly. In classical physics, we just say there are atomic bonds. Atoms exert strong forces on each other that keep two atoms a fixed distance apart. It also keeps angles between atoms fixed.
This makes rigid bodies possible. Very large collections of atoms stick together, where each atom has a fixed place. If you push on an atom at the surface of such and object, the surface atom pushes on its neighbors to keep them the proper distance away and at the proper angle. The neighbor atoms push on their neighbors and so on. The net result is the entire rigid body is pushed without deforming it.
There are also bonds that result in liquids and gasses. And more complicated things like you and me. We are partially solid and partially liquid, and not all that rigid unless we tense our muscles.
When you stand on the floor, you would fall toward the center of the Earth, except that the floor pushes upward on the soles of your feet hard enough to keep you still. When you stand, you are rigid enough that neighboring atoms push on each other and keep all the atoms in you in their proper place. You don't fall to the floor like you would if you relaxed.
When you lift one foot off the floor, the same thing keeps you upright, except that the forces between neighboring atoms are different. All the upward force from the floor is under just one foot. But still, the atoms at the bottom of that foot push upward hard enough to keep their neighbors in their place. Those atoms push on their neighbors and so on. All the atoms in you stay in their places because of forces from neighboring atoms. Now the forces in your uplifted knee are attracting atoms below them so your leg doesn't fall off.
A: 
Let me explain. I understand the Newtonian (classical) physics that
there are equal and opposite forces in play. So when I am standing the
force I am exerting on the ground due to gravity is balanced by an
upward force from the ground.

While this is incidental to the question and its subsequent answer, I shall point this out nonetheless; you have your action/reaction force pairs in error. The force of gravity of the Earth that is pulling you down is 'balanced' by the gravitational force that you exert on the Earth. Via you accelerating into the ground, the ground exerts an upwards force on you, and that force is 'balanced' by you exerting a force on the ground that is equal in magnitude yet opposite in direction to the ground force exerted on you. The gravitational force of the Earth and the force exerted on you by the ground sums up to a net force of zero, as you described, and thus you do not accelerate.
You then go on to ask how the forces 'know' how to shift and move around such that when you lift your left foot, the net forces somehow know how to shift to your right foot such that the net force still sums to zero. Of course, the forces don't 'know' how to do anything, as forces are not conscious. The behavior of forces is axiomatically assumed via Newton's laws of motion; in this case, Newton's third law is pertinent (as per Wikipedia):

When one body exerts a force on a second body, the second body
simultaneously exerts a force equal in magnitude and opposite in
direction on the first body.

Newton's third law absolutely answers your question with utter and complete ease. When you lift your left foot, the forces shift instantaneously so as to produce the results they produce in accordance with the laws of nature. That is just how it is.
Your question is akin to asking, "If I punch you in the face, how does your face 'know' where and how to get damaged? Why doesn't your kneecap get damaged instead? How does the kneecap 'know' it didn't get hit, while your face does 'know' it got hit?"
Such a question is rather absurd, as forces don't 'know' how to behave. Newton's third law is an axiom of which stipulates that that is how forces behave. It is literally just how forces act as per the laws of nature (within our given universe), and there is no 'knowing' on the force's part.
A dissimilar axiom to that of Newton's third could have been one in which the reaction force exerted by object B upon object A, via being acted upon by object A, is applied in the direction of object B's velocity vector crossed with the direction of the force exerted on object B by object A, with the initial force of object A applying in the direction of the ray joining object A and object B.
Such an (absurd) axiom would lead to its own deductive mathematical framework. Whether this axiomatic framework be consistent or not is irrelevant; what matters is how the forces would 'know' how to do this. Answer; they don't know (how could they), for such behavior would merely be axiomatic to the system.
A: I think the confusion starts with the sentence "the force I am exerting on the ground due to gravity".
Gravity might indirectly cause you to exert force on the ground, but it is not the direct reason.
The force you are exerting on the ground is an electromagnetic interaction between the atoms on your feet and those on the ground.
Specifically (and this is where it gets more nuanced and I might not be completely accurate), as the feet atoms come closer to the ground atoms, their electrons are forced into a shared orbital between the foot nucleus and ground nucleus. Since Pauli's exclustion principle prevents two electrons from having the same state, the electrons climb to a higher energy state. The energy gradient manifests as a repulsive force - the feet are pushed upwards, and the ground is pushed downwards.
This repulsive force is greater the closer the atoms are - but it's not a distance you can see. We're talking about the order of magnitude of an atom's size. It could be, for example, that at a 0.2 nanometer distance there is actually electromagnetic attraction, at 0.1 nanometer there is some repulstion, and at 0.05 nanometer there is very strong repulsion. You can't see these distances, but the atoms "know" how close they are to the other atoms.
Taking your body as a whole, you have gravity pulling downwards, and electromagnetic interaction pushing upwards. Equlibrium is achieved when your distance from the ground is such that the repulsive force upwards is exactly balanced by gravity's pull downwards. Then the net force is 0, and there is no acceleration.
(Of course, a more detailed look would consider your individual atoms, and how each is affected both by its own gravity and electromagnetic interaction with surrounding atoms. Your bottom-most atoms are pushed down by the atoms above them, etc.)
When you lift the left foot - at first, the right foot stays where it was, and the same repulsive force is applied. But there is no longer force on the left foot. Meaning the total upwards force on your body is half what it once was. But your gravity stays the same, so you have a net downwards force on your whole body. This causes downwards acceleration for your body, and in particular your right foot. So it goes down and becomes closer to the ground (by a fraction of a nanometer)... Until the repulsive force grows to match gravity again.
It's the same with everything else. Each atom tracks the forces applied to it locally, usually heading towards some sort of equilibrium. Sometimes the effects of a microscopic change propagate, causing a phenomenon that can be observed globally.
A: We've had many answers already, but I'll make one more point I don't think has come up yet. Forces don't have locations; the bodies they act on do. (Forces have directions, however.) It's not that there's a force in each foot at first, and in only the unraised foot later; it's that each foot has a reaction force acting on it at first, and later the raised foot no longer experiences this force, while the reaction force on the other foot increases (albeit not due to some kind of "conservation of force" law, as no such law exists, but that's already been discussed).
Similarly, the gravitational force Earth experiences from the Sun isn't "at" the Earth, constantly moving as it traces its orbit; the force doesn't exist anywhere. This example may actually be easier to reason about, as it only involves two forces, both gravitational; the other is what the Earth exerts on the Sun. (In particular, there's no third reaction force in this example.) Again, neither of these forces is "at" the Earth or Sun, both of which move somewhat; they act on the Earth or Sun.
A: think of it like a piece of code that runs in an infinitesimally small time so we round up all the forces make the object move and then check again so the force doesn't know how to move it is the thing reacting to the new scenario. so think gravity, for example, you put something in the air gravity acts on it but the field is still there acting on the object so the force is with the object, with the normal force it is a reaction to the force you place down on it so if you are on the ground a little less than before the normal force reacts to that lesser force to keep newtons 1st law in tacket. hope this helps let me know if need more detail
A: Assumptions
Energy is conserved
Static forces are all relative potential energy.
Zero motion (macro) is a system in equilibrium.
Change in position required human energy, but net forces (due to F=mg) never changed.
Only the % of total force shared by each foot changed.
Mass did not change
gravity, g did not change.
Human energy was dissipation raising one leg as heat but no sweat.
A: Long story short, the upward force is electromagnetic repulsion from the molecules of the floor. As soon as you raised your leg, you removed it from the floor molecule lattice, so it was not close enough any longer to feel a strong repulsion, like before.
