These are good thoughts above.
What would be useful is if you could state what kind of vehicle and equipment you can afford and plan to use.
Having done that myself many times, I'd propose a minivan with middle and back seats removed and a 150 gal rubbermade tub in it (actually two fit in a dodge grand caravan, if needed) with ~50 or more gal of water and several air pumps (could add a powerhead too) to be run off a power inverter. In May you will not need to heat. The AC in the van will keep you and fish comfortable. There will be plenty of space to stuff other things in the van too.
I'd lightly rinse the biomedia (that sinks; don't want to take up water surface) and as stated above place it in the tub (which is the alternative to the kiddie pool mentioned above). The fish can weather it out a few days in there, if they have to, until you setup your tanks and filters.
Perhaps as Squint says you could use your tanks themselves in place of the tub. I personally would be concerned with blowing seals due to sloshing that can be unpredictable and violent and due to tanks potentially being on uneven support - that can crack the bottom mid-drive. I personally would be more comfortable transporting the tanks empty of water (can stuff them with pillows, linen, clothing, blankets, etc.).
If you wanted a cover on the tub, can use about ten 2" spring clamps ($1 each from Home Depot) and a piece of plastic, what not. Of course, this will not make it slosh-out-proof but will keep the fish and some moisture in and resist a mega slosh-out.
If your vehicle will be small, I'd probably bring all the fish to me (seeing that I run a professional fish rescue

), kidding - try to sell them to adequate homes, sell the tanks and move without all these additional stressors. You would learn a lot from the move but it can be totally exhausting and overwhelming and it WILL add a ton of stress to the move. So this depends on your personality and whether, if worse comes to worst, you can stomach a total loss. No kidding. I was moving 75% of my entire collection of 300+ lbs of fish in a trailer lined with rubber liner from upstate NY to Naples FL in the freezing cold of early April - that's 1500 miles and 40 hours behind the wheel alone with no rest except short 10-15 min sleeping stops when I couldn't keep my eyes open and then keep going. I think I pinched an air hose at some point after a check up and they all suffocated, except two alligator gars.
I did more or less successfully move the rest of my collection, including about 200 koi of 1'-2' in the same trailer and a lot of fish in the rubbermade tubs in my dodge grand caravan.
That's an epic 80 h trip from Washington state to Ohio with 19 large catfish in a box truck with partial water changes on the route etc. from Big Rich:
https://www.monsterfishkeepers.com/forums/threads/big-catfish-need-homes-free-wa-state.359060/