An electric field (as from a charged object) polarizes other matter in its vicinity, resulting in a force attracting the charged object and the surface. To think of it in detail, if I have a positively charged comb, the molecules in the tissue see an electric field nearby. The electrons in the molecules are attracted towards the comb, and the nuclei repelled. They move a little bit apart--not much, but the result is that the like charge is farther away from the comb than the unlike charge. Since the electric field is stronger for the nearer unlike charge (attracting) than for the farther like charge (repulsive), the net effect is an attractive force.
This attraction is even stronger when the surface is a conductor: the comb seems to see a reflection: an oppositely charged comb.
That's electrostatics, though. Which means the motion of the objects (even the little bits of tissue jumping in the air to meet the comb) is much smaller than the speed at which charges reorganize themselves.
A recent study asks what happens when the external charges are moving so fast that the charges in the surface can’t keep up. For instance, what happens if the external charges are moving faster than the speed of light in the wall medium. The author claims that this results in interactions similar to Cerenkov radiation and gives a repulsive force: provided the charges are in a line and the line is going sideways. Or at least roughly so. Points are always attracted.
I don't think this will have much impact on accelerator design, since typically the lines of charge are moving in the same direction they point, so the effect should be minimal. Still, if he hasn't made a mistake, interactions at high speeds start to look quite different from those at low speeds. They do in other cases too.