An active compost pile is heavy, wet, and not something you want sliding around in the back of a moving truck. Deal with the pile first, then move the empty, cleaned bin like any other piece of yard equipment.
Pro tip: If your compost is nearly finished, screen out a bucket of the best material to jump-start your new pile; it is full of beneficial microbes that take months to build up from scratch.
Watch out: Never seal fresh food scraps or an active pile inside a closed bin for the drive; anaerobic decomposition in a hot, sealed container gets genuinely foul within a few hours.