D.B.
I ended up buying a bunch of hard clear plastic storage boxes. You can buy two or three different sizes to help contain things. Find one with easy snap lids. My kids did best with one where the handle helps to lock the lid in place so stuff doesn't fall out when you're carrying them around.
Also collapsible "boxes" (thing kind with the metal spring and mesh lining) worked well for collecting toys that ended up in other parts of the house. Then just make a game where once a week you get everything resorted into the normal containers.
Having the plastic storage boxes also made it easy to rotate the kid's toys. Stash tow or three boxes in the closet and then replace/rotate it with the toys that aren't being played with every other month. The kids tend to play with the toys longer and tired of them less.
Sooo you can one of the pop-up container for the loose pieces down stairs and just keep a bigger plastic upstairs that can hold them all -even though it will be half empty.