A diaper changing station is only as stable as its anti-tip anchor and only as light as what is left in its drawers. Empty it, secure the pad, and it moves like ordinary furniture.
Pro tip: Photograph the anti-tip strap mounting height before removing it, since the correct height depends on the specific stud location, not a standard measurement.
Watch out: Reinstall the anti-tip wall strap at the new home before you put anything on top of the station; without it, a top-heavy loaded changer can tip forward.