After three moves in the last three years, two of them being cross country (ie, more than 500 miles), I've learned a few things about moving your Lego collection that I thought I would share with others so you don't make some of the same mistakes that I made. For every move I've disassembled everything, so this isn't necessarily an article about transporting MOCs, but more for moving your entire collection in general. I'll try to list some of my thoughts as bullet points, so here we go.
1) If at all possible, move your collection yourself. Don't trust anybody else to do it for you.
2) No, seriously, if you can move it yourself, don't let other people pack and move your Lego. You are probably the only one that realizes the value of your collection and the time and effort it takes to keep things organized and the chances of you losing bricks/figures or having it arrive at your new location a huge, jumbled mess are pretty small if you do it yourself.
3) Before you box items, one thing that worked wonders for me this time was plastic wrap. You can get a roll of it for fairly cheap, I got mine at a U-haul rental store. It looks like this LINK. Use this liberally around all your drawer sets and you can even use it to keep lids on or containers from opening up.
4) Box everything. I mean EVERYTHING. Even those big Sterlite drawer sets, and containers with the snap top lids that lock that aren't supposed to open. Even if you used plastic wrap on them and they won't open. That way if anything does come open, it will at least be contained in the box and you probably won't lose anything. I did that this time, and the movers still managed to put my drawer sets vertically into boxes LINK. I didn't lose any parts this time, but I still ended up with a mess like this to clean up once I got it out. If you are having people pack your stuff, it might not be a bad idea to just pack your Lego yourself.
5) Give the movers very clear and very specific instructions on how to handle your collection. On my first move, I wasn't there to supervise when all my stuff was loaded on the truck and the movers just stacked all my Sterlite "shoebox" containers and drawer sets unboxed on the truck. Needless to say when the truck arrived and the door open, the Lego bricks literally spilled out the truck door into the parking lot and I lost a LOT of pieces and figures.
6) For both of my cross-country moves, I had to submit a damage report detailing anything that was broken or lost. If you need to do this, make sure you save receipts. When I moved two years ago, the movers lost/damaged about a half dozen semi-rare Star Wars figures and the guy didn't believe me that it cost me nearly $100 to replace them all (and a few other parts that also were damaged) until I sent him the bricklink invoice.
I think that's about it, if anybody has any additional advice, feel free to post here as well. Hope this helps you out!