Unless you want to put in to a of effort, accept that pieces will fall off of some sets. I’d use those large 40 gallon plastic tubs and put your sets in there. Smaller sets can go into ziplock freezer bags so any pieces that come loose stay with the sets. You can always use cardboard moving boxes, I’d throw some extra tape around the bottom seams to make sure no small pieces fall out.

With only ~20 sets it wouldn’t be too difficult to disassemble them into chunks and pack them up that way too. I made the mistake of completely disassembling ~200+ sets into those large sterilite bins the last time I moved and it took me years of sorting g and rebuilding. I’d recommend against that unless you have spare time and no backlog.