Let say we observe two galaxies, one little bit closer to us than the other one. If we calculate the speed of the farther one does it mean that according to the Hubble law when the closer one reaches the distance where the farther one was it will have increased its velocity to match the velocity at that larger distance?
Broadly speaking, yes.
The "speed" we're talking about in the context of Hubble's Law comes from the expansion of space. This is a simple linear relationship between distance and speed. So objects at the same distance are receding at the same rate. Objects twice as far away, recede twice as fast. This is because further objects have more space between us and them and so more expansion pushing them away.
Hubble's Law is a simple linear law. It assumes that space is expanding at a constant rate of 70 km/sec per mega-parsec. That means that an object that is one mega-parsec distant moves 70 km further away every second. Something that is 2 mP away, moves 140 km. The effect is like compound interest - the more it moves away, the faster it moves so this looks like an acceleration from our PoV.
There is also the proper motion of the galaxy. This is its motion that is independent of the expansion and is just the intrinsic motion of the object relative to us. This has to be added to the Hubble motion and is what pushes the galaxy off the correlation axis on a graph of speed vs. distance.
There's no law of physics making things move at the Hubble speed.
For unknown reasons (possibly as a result of cosmic inflation), the early universe was an extremely homogeneous plasma expanding uniformly outward. If you have, at a particular time, a bunch of matter moving uniformly apart, then it follows from symmetry that it'll still be moving uniformly apart (or together) at a later time, because there's nothing to break the symmetry. If there are small deviations from uniformity (which there were), then the deviations tend to get bigger with time (i.e., matter clumps together), but still at large enough scales things remain uniform.
This is just a result of applying the usual rules of gravity to initial conditions that happen to be symmetric. There's no special force making things uniform. The forces are the same that would apply in any other situation, which is to say, approximately, an attractive inverse-square term plus a repulsive term due to the dark energy.
In the early universe before the dark energy term became significant (or in a universe without the dark energy), recession speeds always decrease because of gravitational attraction. So if we suppose that both of your galaxies are moving exactly with the Hubble flow, then by the time the nearer one reaches the distance of the farther one, its recessional velocity will not only be lower than the farther galaxy's speed was, it will be lower than its own speed was when it was at the closer distance.
In the present era, recession speeds are increasing, so the nearer galaxy will be receding faster when it reaches the farther galaxy's distance. But it won't be moving as fast as the farther galaxy was at that distance; the Hubble parameter is still decreasing.
If the galaxies aren't moving with the Hubble flow, then you need to consider their actual motion. If, like Andromeda and the Milky Way, they're on a collision course, then they'll move toward each other until they collide. Their speed toward each other will increase with time because of ordinary gravitational attraction. The dark energy won't have any significant effect because they're so close together.