If you suddenly move a piece of metal, will that disturb the free electron density? If we have a hollow pipe sitting at rest filled with gas and we moved the pipe suddenly along its length to the right, then the gas density will be momentarily higher near the rear of the pipe and lower near its front because gas molecules have inertia. If we start rotating the pipe then gas molecules near the pipe wall will have maximum speed and the speed will go down as we approach the axis of the pipe.
What if this pipe was a long metallic cylinder. Metals have free electrons. Will the free electron density change in a similar way if we translate the cylinder suddenly?
What if we rotate the cylinder? how the free electrons will respond to that motion?
What if the metallic cylinder was charged (so now there is charge resides on its outer surface). How the charge on the surface of the cylinder respond to the sudden translation or rotation of the cylinder?
(And what theoretical model(s) one should consider to answer the previous questions?)
 A: In principle yes, but the electrons will respond at around their natural frequency of oscillation. This is the plasma frequency and for most metals is around the frequency of visible light or about $10^{14}$ Hz. So the electrons will only be displaced for a few fractions of a picosecond.
The analogy with sound is that the motion creates a sound wave that travels up the pipe at the speed of sound. In a metal you would create a plasma oscillation or Langmuir wave that propagates at around a tenth of the speed of light.
A: I think we can't compare a hollow pipe (filled with gas) to that of solid conductor. Gas molecules/atoms are free to move. In solids kernels aren't free to move, though conduction electrons are free to move. Drift velocity of free electrons will be very less (even during current flow). So, we can't compare motion of gaseous atom or electrons with that of electrons in order to account for the same results as you explained. Yes, as explained by john Rennie sir there will be displacement for fractions of a picosecond. So, there will be certain addition to the kinetic energy of electrons (but not the extent as for gaseous molecules).  
However, according to free electron model, on a scale much larger than the inter atomic distance a solid can be viewed as an aggregate of a negatively charged
plasma of the free electron gas and a positively charged background of atomic cores.
A: I think this is in nature a problem of relativity theory.
Since we(observers) take the metal as a reference frame to perform experimental measurements on electrons, then according to the equivalence principle of general relativity, the effects of 'suddenly move' is physically equivalent to gravity. But as we know, the electrons are very light, and usual gravity force can be ignored as compared with, for example, electromagnetic force. Thus, to make a observable effect on electrons, we must generate a huge acceleration and a finite time-interval for 'suddenly move'. But at the same time, special relativity may enforce an upper limit for the acceleration$\times$ finite time-interval.
In conclusion, I think whether a 'suddenly move' could qualitatively affect the electrons depend on the competition of the following scales: speed of light, rest mass of electrons, and (for example) electromagnetic force.
