How do you make a player move on a remote items local axis?

Doing a 3rd person game where you can aim around in game, which will rotate your character. So far I've gotten the Horizontal inputs spin an empty object which has the camera as a child(causing a panning effect around the player, opening new fields of view). The only problem is, it still moves in a world axis, and if I make the player move one /its/ local axis, then when you're aiming with the mouse he moves all wonky and can bug out. So I'd like to make him move on this empty objects local axis instead. (I already have the transform plugged in as a public variable, just unsure how to use it for this particular method)