A plastic drawer unit is light, but its drawers are not locked in place, and a drawer sliding open mid-carry is the most predictable way its contents end up all over the floor.
Pro tip: Stretch wrap alone often holds drawers shut well enough for a local move, saving you the time of fully emptying and reboxing everything.
Watch out: Never carry a plastic drawer unit by a single drawer handle; the handles are not designed to bear the weight of the whole unit and can snap off.