As many of you know, I recently went through the experience of having to move somewhat cross-country (600+ miles). I packed some of my Lego, the stuff I knew would be most probably to make a mess or be spilled, and brought it with me when I drove myself here a few weeks ago. The bulk of my collection, stored in snap-top totes went with the movers. Unfortunately I had to start work and was not in Michigan when they loaded up all my stuff last week, so I was not there to supervise how my collection was packed up.
They did not put my snap-top containers into boxes, and several of them tipped, or were jostled around during transit and spilled their contents, which surprised me. As they were unloading things this morning, there were tons of bricks all over the bottom of the truck, and some of them ended up in the street. The movers, trying not to let me see how much damage was done then started randomly shoving bricks into any container they could find, which means I had to re-sort through everything to figure out what I still have and what is missing. So far I have roughly 80 1x4 bricks in various colors that were severely damaged, several arches were badly damaged, 3 Star Wars minifigs are missing helmets, 2 are MIA, and one Tusken Raider was mauled in the face. I still need to go take inventory (which will take weeks) to find out what else is missing.
Anyway, to make a long story short, if you plan on moving, I would highly recommend putting EVERYTHING in cardboard boxes, even if it's already in a snap-top tote. That way it seems like your chances of things tipping or spilling is significantly less, and if it does spill in the box, at least it's still somewhat contained. Or, just move everything yourself if that's possible.