A changing table is essentially a small dresser with a raised top or a removable topper. Handling the topper and the drawers is all it takes to move it cleanly.
Pro tip: Pack the changing pad and diaper supplies in a clearly marked box so the changing station is easy to set up first at the new home.
Watch out: Do not carry a changing table by the raised top rail, since it is often just screwed on and can pull loose.