Propagation of static electric field

At what speed does a change in a static electric field propagate through a medium? I understand that through a vacuum it's the speed of light, but through a medium such as a conductor or insulator how fast will this change propagate? In classical electrodynamics, how do we model
the propagation of this change - as an electromagnetic wave? If so, is there anything that
can slow or absorb the propagation of this electromagnetic wave carrying the information of the change?

For a concrete example, suppose I move a charge on one side of a wall to a new position on the same side of the wall. On the other side of the wall I wish to determine how long it takes for the information about the change in charge position to be a detectable change in the electric field
on the other side of the wall. I imagine that any speed of propagation different than the speed
of light would be determined by the wall medium and the frequency of the electromagnetic wave propagating the change? If the acceleration of the charge during the move is small, does that
mean a low frequency EM wave which would not normally be slowed by the wall at all?

Also, a related question which probably draws in quantum electrodynamics - once the charged
particle is moved and remains in place at its current position for some time, how is it that the
space on the other side of the wall contains the information about the static electric field there?