A lot depends on how you move. You can put anything in a Uhaul. I had a lot of tall plants I put in last into the truck just wedged in to keep upright. We opened the back when we stopped to let cool air and light in for a moment. We were towing a car, so the smaller plants went in there packed in a shallow box to keep the car clean and the plants upright. Be advised that if you move to CA, you are not to bring plant material in. I am not sure about other states. I am not sure what a moving co. would do. This might just be your opportunity to start fresh with plants too, which is not such a bad thing. Good luck.

big movers don't take plants. not even for a few miles move (know the hard way) - unless they are a small mom'n'pop shop. then may be, but you need to advise them how to pack/stack them.
have a large assortment of small to large cardboard boxes. set the pot in plastic bag, tie off around the pot, not too tight, just as close to crown/trunk as possible. do not water just before the move, but sev days before for large plants.
you don't want the cardboard to get wet and mildew.
then start with smaller plants - put them in the smallest box their pot fit. good to put sev together. then you can wedge the smaller boxes into a large box with big plant. if bottom pots/boxes fit tight you don't need to do anything more.
you might want to drape large dry-cleaners bags around foliage to keep it from spreading/tearing.
leave tops open, may be raise flaps and tape in up position to prevent anything falling/shifting from the side. you can build a tall protection box this way for tall narrow plants.
generally it's best to put plant boxes as top/last tier loading: and make sure nothing can shift and topple into them.
or you can just wedge in empty cardboard boxes (taped shut for strength) - like from amazon shipments : flat like bricks, inside the box, around the pot of a large heavy plant, completely wedging it.
the tall dish packs are sturdiest for large plants, but it might be hard to drop the plant in: they're 3' tall?
the large square boxes are flimsy, but easy to stuff in the back of the uhaul. big sturdy plastic storage tubs (square, stackable) are great. also you can use 'dirty laundry tubs' - they are big and sturdy.
also you might use the small uhaul van - if plants fit height-wise in one layer.. that's the easiest to drive/pack.
if driving overnight, open up to air, give them some light, check,etc.
for short move no big precautions are necessary. for sev days move you'll need to punch holes in sides of boxes, if you need to close them and tape them (for small plants).
best not to drive in very hot sunny weather - they can cook!
you might need to drive only in the morning/evening.
curious about the cali drive - that would be very hot most likely? and sev days. some plants might need to be watered ..
but since cali is quarantined (? didn't know about that!) - taking plants is not an option then.
though even with quarantine you might be able to take some plants, you'll need to check what the proper procedures of entry/inspection are. like in fl/hawaii some nurseries provide 'certification' services for shipping. you'll need to pre-certify it then.

Please do your own research about moving with the houseplants. You'll find that you CAN move plants into California with certain limitations. They have to be packed to that they are readily accessed if inspections are required.

Anyway, I've attached a link to the California Department of Food and Agriculture website for further information. You should do the same kind of web search before packing to move to any state. You will need to comply with all state and federal regulations concerning the transport of plants, soil, even garden produce across state lines.

Some states are very cautious!

Here is a link that might be useful: Click for information

Thank you everyone for the insight. It is looking more and more like a reality California is the way we are heading. So that means flights and long travels. Thank you Rhizo for the link. I will look into it closer when I get home.