1. Make sure you have a good inventory
Part of having a moving company haul your stuff from one place to another is that if anything goes missing, you have to prove it's missing. Having an inventory of what you have is important for tracking. For the business side, I already had a list of the items I had to include minifigures and unopened sets so that was easy. For the personal collection side, not so much. I decided to take the risk on loose bricks (business or personal) and no inventory them so that make it a bit easier. If you are a Bricklink seller, then you might already have an inventory of your bricks and this won't be an issue.
2. Take pictures
Another step to proving you own it is taking pictures. This is what I did with a lot of the sets in my personal collection. For some of the smaller sets, I just took a picture of the large box they were all in, but for some of the larger sets I took pictures of the set separately or in the case of modular buildings, pictures of each floor. For the business side, I took pictures of the conditions of the unopened sets that I am selling. The last thing I want is a damaged modular building or Star Wars set. If the packers or movers destroyed an unopened box and I have a picture showing that it was good before it was packed, I should be able to claim it with the moving company.
3. Use boxes from LEGO® where you can
Some sets are oddly sized and movers come with specific sized boxes that may not line up. I placed some of the larger boxed sets that I had in some boxes that I had saved from the LEGO® company. This made packing easy. For those that have never had movers come in and pack their stuff, you are supposed to let them pack everything. That means you can't use your own packing material and tape boxes shut and then claim damages after they are shipped. I placed sets in the boxes I had and then let the movers tape them shut.
4. Bag up loose and sorted bricks ahead of time
When the packers showed up at my house they just start opening drawers and putting stuff in boxes. I can't imagine what would have happened if they had to sort through my loose bricks that are separated by type and color. I took the time ahead of the packers arriving to dump my sorted bricks into plastic bags. The bad part is that I need to replace them in all of the drawers that I had, but the good news is that I don't have to re-sort all of my bricks.
5. Transport anything yourself that you just can't bare to lose
If you have a set, piece, minifigure, MOC, or whatever the case may be and you don't want to lose it, transport it yourself. During this move I decided to let the movers take everything. In a past move, I had an unopened Emerald Night set that I decided to transport on my own. During this move I decided to let the movers take everything. What would you want to transport yourself? Do you have an unopened Ultimate Collector's Millennium Falcon (#10179)?