What causes displacement? I'm doing this physics question, which I'll paraphrase:

A car being driven at a constant speed 20 m/s in a straight line goes through a school zone. The car passes a tree, behind which a police car is waiting at rest. After five seconds of passing, the police car accelerates after it, parallel to the path of the other car, at 2 m/s². At the instant when both cars are traveling with the same speed, what are their positions (relative to tree), and why (are they different)?

So I was answering the "why" part of the question, and I thought that the positions of the two cars at that instant in time is different because their respective displacements from the tree are different.
Then I wondered: what caused the displacements to change? It could be the velocity, but I thought that since velocity was derived from a function of displacement over time, it didn't really make sense to me that velocity causes displacement to change, but rather that it was a description of how displacement changes.
More broadly, what causes things to move (with or without nonzero acceleration)?
I've always heard that net Forces cause acceleration, but how does acceleration cause displacement to change?
Edit: I just realized that the same thing can be asked of the relationship between position and displacement.
Edit: Concerning Mauro Giliberti's and Cinaed Simson's comments. If displacement is so connected with velocity that there is a iff relationship, then my question is this: what changes the location of the object as displacement, velocity, etc. is changed?
In the main body of my question, my use of "displacement" involved both the real-life sense (where the cars were and are) and the non-real-life sense (numbers and values). In this edit exclusively, I have gone along with Mauro's and Cinaed's use of "displacement" (if I'm not mistaken) in the purely non-real-life sense. I don't know this whole concept well enough to rephrase "cause" so I hope this edit was adequate.
 A: I think the best description that can be had is that motion is representative of a truly fundamental process in physics that, as far as we know, does not admit further reduction into more elementary processes.
If one takes the reasonable understanding that an interaction is anything which causes a change in the physical state of an object, then given that part of the total physical state includes the object's position, changes in position must also be considered as the result of such an interaction: we can say that every physical object always undergoes at least one self-interaction, which converts the momentum of the particle, another part of its state, steadily into displacements. This is the interaction that generates motion, and it should be mentioned as such alongside other types of elementary interactions.
However, asking "what this is", beyond that, is not a question physics can answer, any more than it can answer what the other types of interactions "are" beyond giving us descriptions of them, e.g. while we can talk of charged objects interacting with an electric field, say, only now to cause changes in their momentum, we cannot say what that interaction "is" any more than stating the fact of its existence and describing just how that it affects the charged object's physical state. Or to put it another way, we do not have access to the Universe's "source code", so to speak, so we don't get to see how anything is actually implemented "under the hood" :D
A: The question you ask can be rephrased as "what causes motion," which makes it more clear that this is a deep philosophical question.  Science doesn't actually have an answer.  In science, we observe that motion happens, and try to come up with models that predict the results of that motion.
Velocity does not cause motion, as much as it is defined to be a mathematical way of approaching changes in position/displacement over time.
Later in your science career (much later!) you may come across Lagrangian mechanics.  Lagrangian mechanics is a complete reformulation of Netwonian mechanics.  It yields exactly the same results, except instead of capturing motion with $F=ma$, it is captured with a very different concept, called the action, $S[q]=\int L\ dt$ and defines the true motion of the system to be the path which makes $S$ stationary (minimized, maximized, at an inflection point).  This is a completely different way of formulating things which points towards the concept of motion as a "minimizing" agent (or at least stationary).  Yet it reveals the same results.
Regardless of whether you are using Newtonian mechanics, Lagrangian mechanics, relativistic mechanics, or quantum mechanics, we have a concept of motion which is based in our intuitive concept that things "move."  Trying to separate the mover from the movement is like trying to separate the doughnut from the hole.  Philosophers will revel in the distinction between them.  Scientists will take it as a whole, say "don't mind if I do," and eat said doughnut.
Then they'll eat 12 more of them, for statistical rigor.
A: $N_1: m\frac{d^2}{dt^2}(x) = F_{res} => \frac{d^2x}{dt^2} = \frac{F_{res}}{m} => x(t) = \frac{F_{res}t^2}{2m} + C_1t + C_2$
Here $C_1$ has dimension $\frac{m}{s}$ and $C_2$ has dimension $m$. Thus we can see that the magnitude of the resulting force, as well as the amount of time a resulting force acts on an object, determines the position at time $t$. Also, initial velocity $C_1$ determines the displacement, but $C_2$ does not determine displacement as displacement is: $\Delta x = x(t+\Delta t) - x(t)$, and thus the $C_2$ term is removed from the problem. Keep in mind that a force must act on an object for some time to change its velocity significantly: $\frac{d\vec{p}}{dt}=\vec{F} => d\vec{p} = \vec{F}dt$.
A: Instantaneous velocity and displacement, although related by a differential equation, are independent attributes of an object. Neither one can be derived from the other. To derive displacement from velocity you need to know how velocity changes over a period of time, not just velocity at one instant. Similarly, to derive velocity from displacement you need to know how displacement changes over a period of time.
So just because the cars have the same velocity at one instant, this is no reason to think that their instantaneous displacements should be the same. Indeed, it tells us nothing at all about their instantaneous displacements.
Asking “why are the cars’ displacements different if they have the same velocity ?” is like asking “why are the cars different colours if they have the same velocity ?”.
A: 
Then I thought what caused the displacements to change? It could be the velocity ...

It's not the velocity. Velocity is synonymous with change of displacement.

More broadly, what causes things to move.

This will depend on the physical system. For example, if you picked up a stone and thrn let go, it will be the earths gravity that causes the stone to fall.
If you got into a rocket and then set off for an earth orbit, then it will be the rockets thrusters that cause the motion. Of course, here, it is the ejection of hot gases that cause the forward motion by conservation of linear momentum. Nevertheless, this wouldn't occur without the thrusters on the rocket.
At bottom, we don't know causes motion, it's an irreducible physical phenomena.
A: Your question “what causes displacements to change?” is basically the same as “what causes velocity?” since the definition of velocity is change in displacement per unit time when the displacement is measured from a fixed origin (rather than the displacement that occurred during that last second which would also be a fair way to think of it). Actually there may be some confusion here about terminology which I have never seen mentioned before. The definition v = dx / dt normally we take x as displacement, but we could also think of dx as a  displacement.
Anyway in physics net force causes acceleration or change in velocity. But velocity itself doesn’t need a cause as such. In free space objects just whizz along with constant speed and direction with no evident cause or hand needed to guide them other than spacetime. Once something is moving no further cause is needed to keep it moving. Like a rolling ball. Note you must beware of frictional and air resistance forces which you might not have considered in some situations. Anyway the tendency for things to just keep moving with constant speed and direction is  Newton’s law of inertia (which also includes why a stationary body stays stationary if no one pushes it). We could call the “cause” inertia but that is no more an explanation than a recognition of a physical law.. that is just how it is and is considered basic so can’t really be explained in terms of anything more basic. People may try to get deeper but that would be just philosophy at the moment.
A: There are two possible answers:
displacement is the cause of the change in position. This is a tautology: displacement and position, in this analisys, are the same thing. You can say that an object changes color because it changes hue, but that wouldn't be much of an explanation would it?
velocity is the cause of the change in position. You claim this could be a problem because the velocity can be expressed as a function of position, but that doesn't mean anything! Physical formulas are (often) invertible, while effect-cause relationships (often) aren't. Philosophers of science are still debating if physics implies the direction of an effect-cause relationship or not (see Time and Space by B. Dainton), but I think we can be pretty sure about the fact that it implies the presence of such a relationship. You say that force causes acceleration because there is a formula, $F=ma$, that links the two and one can't happen without the other. There are other formulas with acceleration, and the quantities of the other formulas may also be regarded as "causes" but this is the simplest it can get. Analogously, I claim that velocity causes a change in position because there is a formula, $v=\Delta s / \Delta t $ that links the two and one can't happen without the other.
A: It might be helpful to "Think iteratively", borrowing a phrase from Chabay and Sherwood's Matter and Interactions ( https://matterandinteractions.org/ )
"Thinking Iteratively" (AAPT Meeting 2014)
 https://youtu.be/e-shsRZQsi4?t=64
 https://youtu.be/e-shsRZQsi4?t=683
Here is an example of code in Glowscript/VPython ( https://glowscript.org/ ) that "evolves the motion of a binary star system".
From https://glowscript.org/#/user/GlowScriptDemos/folder/Examples/program/BinaryStar-VPython
the relevant portion of the code is
while True:
    rate(200)
    r = dwarf.pos - giant.pos
    F = G * giant.mass * dwarf.mass * r.hat / mag(r)**2
    giant.p = giant.p + F*dt
    dwarf.p = dwarf.p - F*dt
    giant.pos = giant.pos + (giant.p/giant.mass) * dt
    dwarf.pos = dwarf.pos + (dwarf.p/dwarf.mass) * dt

In words,

*

*In a short increment of time $\Delta t$, the net force on the giant star provides an impulse, and
thus increments the momentum of the giant star:
$$\vec p_{new}=\vec p_{old}+\vec F_{net}\Delta t$$

*During that short increment of time, the updated velocity (as momentum/mass) provides a displacement,
and thus increments the position of the giant star:
$$\vec r_{new}=\vec r_{old}+\left(\frac{1}{m}\vec p_{new}\right) \Delta t$$
