For every risk you have three options: Accept it, insure against it, or mitigate it.

Make sure you understand how insurance applies to things being moved by your moving company, both your insurance and whatever insurance they provide. Insurance is not typically a good way to handle records that are lost, stolen or destroyed.

I'm a big fan of backups. Almost all of my important records are digital and backed up. Anything that can't reasonably be backed up should probably travel with you if you're traveling by ground. If you're flying and it won't go in a carry-on, I would send it FedEx -- in my opinion they are the most reliable package and document carrier (no affiliation).