Storage bins are already built for moving, which is exactly why they get overloaded and mislabeled. Tape the lids and label both sides and they become the easiest items in the whole move to track.
Pro tip: Color-code bins by room with colored tape or stickers for a system that is faster to sort at a glance than reading labels on every single bin.
Watch out: Do not overfill a bin just because it has room left; an overweight plastic bin can crack at the corners or the lid can pop under the strain.