I've moved several times, as I spent 20+ years in the US Air Force. When those times came, similar to what jimq said, all my lesser cards (those in binders) went with the moving van. I put all my higher value graded stuff in the trunk of my car, and in the back seat covered with miscellaneous stuff and blankets. I would only stop for gas, and bought drinks and snacks I could eat in the car as I drove. When I'd stop for the night on a multi-day drive, I'd always stop at an Air Force base and get a room in base lodging.
The only issues I ever had was once when the movers decided to use my TV/stereo wall unit as a "ladder", and broke the top shelf of it when one of 'em climbed on top of it
The other time was when a 3'x3'x3' box came up missing upon delivery. A week or so later, I got a phone call from the movers (a husband/wife team) telling me they had found the box during a subsequent delivery. They mailed the box to me a few days later (it contained kitchenware/pots & pans).
Steve