Does a classic physics describes the electric charges forces propagation speed? When you launch this applet you can notice, that in the beginning the force lines are propagating from a charge, with some speed(speed of light, probably).
The force lines means, that in every point of the space there are force with straight-defined direction, according to charge position. When charge is changing it's position, the force direction and density should change too.
So, how does a classic physics describes the electric charges forces propagation speed?

 A: Considering Maxwell's equations as "classical" physics, the propagation speed of a change in the electric field strength can be calculated from those equations. Maxwell himself did this and discovered that this speed was equal to the speed of light. 
A: EM interactions are mediated by photons. Photons are massless particles that travel at speed c in vacuum when measured locally. All massless particles travel at speed c, like gluons and theoretically gravitons.
Originally in classical physics Maxwell's equations described the classical EM waves and their propagation. In that theory, a changing E field induced a changing M field, and a changing M field induced an E field, and this is how classical EM waves propagate.
Now these equations of Maxwell have a solution for the propagation speed of the classical EM waves, and that speed is c, in vacuum, when measured locally. In the classical theory, this speed has a relation to the permeability and permissivity of vacuum.
Nowadays, the speed of light is defined exactly as 299,792,458 m/s. It is a universal constant, and its value is exact, because the meter is defined as the distance traveled by light in 1/299,792,458 seconds.
Today, we know that not only EM waves, and photons travel at this speed, but all massless particles, and gravitational waves too.
When you say force lines, and charge position, if you talk about an electron, a charged particle, that has rest mass, and cannot travel at this speed. Nothing, no macro object and no elementary particle with rest mass can travel at this speed.
Photons do not carry charge. Now the QM description of EM waves is a herd of photons traveling as waves, and the classical EM wave works good with the QM theory.
If you are talking about charges traveling, like electrons, in a metal wire, the drift speed of electrons, to travel from one atom to another one, is slow. But, since, the electrons in the metal are so densely packed, the EM charge travels inside the metal with speed near c, that is why your applet is relativistic.
You are asking about force fields and based on the correct comments:
Let's take an electric charge like the elementary charge, the electron. It has field lines around it, those are mediated by virtual photons. We call them virtual because they describe the math. They too do not have to obey some laws that real photons have to obey, for example they are not limited by the speed of light. As the electron travels in space, the force field travels with it. The direction of the force field does not change, relative to the electron, the force field lines look constant.
