How does the magnetic force exerted to moving electrons in a current-carrying wire, actually transmit to the wire itself? From my physics book I understand that the magnitude of the magnetic force is $F=qvB\sin(\phi)$. It is posteriously derived that the total magnetic force exerted to all electrons in a current-carrying straight conductor is $F=ILB\sin(\phi)$. But how is the force transmitted to the wire itself? What interactions are happening within its interior so that there is a net force acting on it?
The doubt arose since the previous expressions are for forces acting on the moving charges. I have some notion of the Hall effect, but the related electric field just compensates for the magnetic force on the electrons, and does not represent the interaction between the nuclei and the electrons.
 A: 
But how is the force transmitted to the wire itself? What interactions are happening within its interior so that there is a net force acting on it?

In short, the conducting electrons push on the surface charges, and these push the conductor via EM and possibly via non-EM forces.
When a conducting moving electron (or "element of conducting charge" in the macroscopic theory) suddenly starts experiencing an external magnetic force, it begins to change its velocity. Thus the average conduction velocity there changes direction, from being in direction of the total electric field, to a slightly different direction, due to presence of the external magnetic field. This happens to all conducting electrons experiencing the magnetic field there.
However, and this is the main point, the affected conducting electrons near the surface can't just move in the new direction out of the conductor; they are prevented from doing so by constraint forces due to the conductor acting back on the surface electrons (unless there is a field emission or other kind of charge emission which overcomes these constraint forces).
In other words, in general, the electrons can only move freely inside and on the surface, but can't jump out of the conductor into the vacuum. Electrons clump on some conductor surface patches (such patches become negative), and they are lacking on other surface patches (such patches become positive).
The constraint force on the surface charge is a net result of all microscopic EM interactions (and possibly non-EM interactions) there between the conducting electrons and the rest. These interactions create something like a steep potential barrier on the conductor surface, which the electron can overcome only if it has high enough energy, or some external force is strong enough. One way to understand this constraint force is to think of what happens when a charged test particle approaches a large body made of many positive and negative particles. As the test particle approaches the body, the attracting force increases (due to polarization of the neutral body), and becomes very high when the test particle is on the surface, as close to the other particles as is the typical particles separation distance in solid body. When the test particle continues and comes inside, the force as function of position becomes oscillatory on a very short length scale, which has similar effect on macroscopic conduction as being zero. That's why we see effects of these interactions and the macroscopic constraint force only on the surface.
Due to changed trajectories and constraint forces, more electrons than before concentrate on one side of the conductor surface, and they move out of the opposite surface. Eventually (very quickly) there is enough of this surface electron separation that it produces electric force inside the conductor which cancels out the external magnetic force at every point of the conductor, and this cancels out further concentration of the surface electrons.
So very quickly an equilibrium is formed, where net magnetic force on a current element is cancelled by a force due to the conductor surface charge acting on the current element.
But this force is electric force, and more or less obeys the principle of action and reaction. So there is also the electric force equal in magnitude and opposite in direction, which acts due to the conducting electrons inside, on the surface charge on the conductor surface. And since this charge cannot move outside the conductor, this surface charge acts on the conductor body and thus produces the macroscopic force on the current-carrying conductor. So somewhat surprisingly, the macroscopic magnetic force $BIL\sin \theta$ is actually a net force due to surface charge and conducting electrons acting with forces on the conductor body, and this is at least partially sum of inter-particle electric forces (contribution of non-EM forces can't be ruled out macroscopically).
