Where does movement come from? When you put a body in motion, where does that movement come from? Ok, you will say things like "acceleration", but where does that acceleration come from? Then you might say "a force is creating the acceleration", but how does the force create the acceleration? So ultimately, where does movement come from?
L.E. I don't want the answer to be restricted to Newtonian Mechanics. I just want to know if there exists in Physics an explanation for where does movement come from. Maybe Quantum Mechanics has some answers? Like a direct collapse of the wavefunction on a specific value for velocity at the exchange of virtual photons in the case of electromagnetic interaction (the direct push of the object for example)?
 A: Physics is the discipline that studies natural phenomena, and finds mathematical models that fit the measurements and observations and also predict future behavior of the system under study.
Mathematics is a discipline which studies numbers with sophisiticated methods, it has axioms and theorems and can prove statements or disprove them absolutely. To use mathematics for observations of nature there should be statements, called postulates or laws, which connect observations with mathematical formulas, one can think of them as "physics axioms".
The behavior of objects under motion, the concept of forces etc. have the postulated three laws of motion of Newton, which are used to develop the elegant mathematical theory of classical mechanics.
So the answer to

So ultimately, where does movement come from ?

is : from observations that we define as "motion" axiomatically. We observe motion and have described it mathematically with classical mechanics.
A: One approach to exploring this question is to study the Glider in Conway's Game of Life.

Where does its movement "come from"? It is a direct and easily-verified result of the rules of its universe. I highly recommend you take a few minutes with graph paper to verify its behaviour for yourself.
What is really important to see is that motion is not a law of the Game of Life universe, nor is velocity a fundamental property of entities in this universe. They emerge from the very simple rules that govern the Game of Life universe.
One can ask, are successive generations of this collection of dots really part of the same entity? Some of the dots making it up disappear, and some new dots appear out of nowhere. But think of the human body. Our cells are constantly dying being replaced. Our bodies are mostly made of water, and any given molecule could have been outside our bodies a day ago. Yet we consider a human an entity.
Note that there are other known patterns in Conway's Game of Life which travel at different speeds (known as spaceships). It is possible for certain collisions to transform a slow spaceship into a fast one, or vice versa.
This can give us some intuition for what forces are. When one entity applies a force to another, it is changing some internal property or configuration of the other entity, and in turn that entity will then move at a different speed (according to the rules of the universe).
For completeness I should say that our universe does not follow the same rules as Conway's Game of Life. However, hopefully I have shown how movement can arise in a system which evolves by a fixed set of rules.
A: At a basic level, current models of physics describe a force as the exchange of (virtual) bosons. This is just a model, but here's the general gist at what's a very basic level:
There are different types of fundamental forces:
Strong, electroweak (electromagnetic + weak) and gravitational. On top of this, there is the Higgs Boson.


*

*The strong force is mediated by gluons

*The electromagnetic force is mediated by photons

*The weak force is mediated by the $W^+$, $W^-$ and $Z^0$ bosons

*gravitational force is mediated by the gravition, which has not actually been discovered, but is believed to exist
The Higgs, in this case, gives mass to the weak bosons by the process known as symmetry breaking.
'The standard model' in quantum field theory (QFT) describes how these (with the exception of the graviton) interact with each other and the other fundamental particles (fermions). Describing an 'object' as some sort of entangled wavefunction of the smaller objects in a larger Hilbert space:
To simplify the explanation, assume that there are only two objects in the universe which (to 'explain' the electromagnetic interaction) both have a net charge, with a negligible mass and are far enough apart that the other interactions have a negligible chance of occurring relative to having an electromagnetic interaction. Let's also assume that they are somehow observed at regular intervals (so that these objects actually have a defined position, albeit with some uncertainty) and that at some time, $t_0$, they are stationary.
What then happens is that these objects 'exchange a virtual photon' (or have some other valid QED interaction), which really means that the maths given by QED give the right solutions to the probabilities, strengths etc. of interactions (really, this is boiling down into physics is just a model - see the answer given by anna v...). What seems to be described by this maths is that one of the objects emits a photon with some momentum and so, by the 'law' of conservation of energy-momentum, experiences a change in momentum. This photon interacts with (gets absorbed by) the other object, which therefore also experiences a change in momentum. In other words: the objects are now moving relative to each other! Having said all this, I need to stress: 'Exchanging virtual bosons' is really physics shorthand for 'the result you are looking for is given by performing the relevant mathematical perturbative expansion in QFT'. As this is difficult for everyday objects (read: practically impossible), we just use classical physics instead. Which, for everyday objects, is good enough.
The other interactions work by the same principle, but there are differences (strength of interaction, distance over which the interaction will realistically occur etc.) and relaxing any of the above assumptions means that other interactions also need to be included...
However, we have yet to include gravity. Suffice to say that the path (and hence, 'movement') of a particle is described by a world line, which, in the absence of other forces, is a geodesic
I've written all of the above as this is hopefully what you're looking for and so, will satisfy your curiosity, but what's important is:


*

*The law of conservation of energy-momentum means that when two or more objects interact (so that they are not in equilibrium with each other), they experience a change in momentum, which is observed as either movement, change in mass, or some mixture of the two. Once an object has started moving relative to another object and it's no longer interacting with anything and its mass/energy is not changing, it stays at that velocity

*All of the above is a model that fits an observation. Again, see the answer given by anna v

*Whether or not it's due to collapse of the wavefunction, decoherence or a number of other possibilities, is down to whichever interpretation of quantum mechanics you're using, which all give the same experimental results (at least, the valid ones do anyway)
A: The movement comes from push, or pull. A push, or pull comes from one of the forces. For example, a push happens when two things/fields can not co-exist at same point of space time and one (or both) of them get displaced. The displacement, is the movement. The forces govern how things behave. Movement is one of the behaviors caused/governed by forces.
