Take three large--very large--plastic garbage bags and cut them down the sides, leaving the bottoms intact to make long plastic strips. Then use extra-wide masking tape to fasten the bags, one to the back cushion of the rear seat, one to the seat cushion, and one to the floor below the seat. If necessary, another long bag can be fastened to the front passenger seat, back and seat, and a half one to the front passenger floorboard. That way, the mud will not stain the upholstery and cleanup requires merely detaching and removing the cut bags. It's understandable that children other than your own might feel a bit ambivalent about riding around in their underwear, although in fact, as you note, they're well covered--social mores don't have to make sense, they only happen to be.