In a paper from Cadwell Magnetic damping: Analysis of an eddy current brake using an airtrack about eddy current brakes the author explains the effect qualitatively as follows:
When a nonferromagnetic conductor passes between the poles of a magnet, an electric field is induced, and circulating currents called eddy currents are generated. As a result, a magnetic damping force is induced on the eddy currents which opposes the motion of the conductor.
As an aluminium plate enters an external magnetic field [...] a Lorentz force [...] is exerted on the conduction electrons in the aluminium plate[...] The velocity of the plate $v$ and the magnetic field $B$ are orthogonal to one another. An induced current moves along a closed path [...].
A horizontal magnetic force is exerted on the portion of the eddy current that is within the magnetic field. This force is transmitted to the aluminium plate, and is the retarding force associated with the magnetic braking.
The part about the moving electrons and the Lorentz-force is pretty clear. But how and why is the force transmitted to the aluminium plate? I mean we are speaking of conduction electrons, so I just would expect that the Lorentz force alters the path of the electrons, but why does the Lorentz force on those conduction electrons affect the aluminium plate as a whole?
And why is the Lorentz-Force which is due to the motion of the aluminium plate with respect to the magnetic field not transferred to the plate as a whole (resulting in a force orthogonal to the direction of motion on the whole plate)?
Here is an illustration of the set up from the paper: