Familiars may only perform one move action on their turns. If a familiar can do anything else, then it can also do that, before, during, or after its move action.
Descent 2nd Edition rulebook, Page 17, Familiars wrote:
When activated, a familiar may perform a move action following the same rules as heroes. Familiars treat any special terrain (other than obstacles) as water during their movement. The familiar may perform additional types of actions during its activation, as noted on its Familiar card. These additional actions, unless otherwise noted, may be performed in addition to the move action and may interrupt the move action in a way similar to hero move actions. If no such other actions are noted, then all the familiar can do on its turn is move up to its Speed value.