We study the electrophoresis of surface-charged thermosensitive microgel particles based on poly-N-isopropylacrylamide (PNIPAM); these deswell with increasing temperature T. Our results show that the electrophoretic mobility mu is affected by the temperature-induced volume phase transition. It increases with increasing temperature, as a result of the charge density increase induced by particle deswelling. Temperature thus allows control of mu, in contrast to the more conventional charged hard spheres for which mu is T independent. Salt also affects the mu behavior and gives rise to rich phenomenology, sharing common characteristics with charged hard spheres and polyelectrolyte-coated colloids depending on whether the microgels are swollen or deswollen. We interpret the effects of salt concentration n by considering that particle charges are located in an external shell, as confirmed by titrations, and that it is this shell-salt-induced compression that affects the resulting mu behavior.