Firstly, let me say I have not hauled long distance and I have not hauled in the USA (but that doesn't shut me down
). Based on the adventures I have heard from other folks hauling their animals, I've picked up this much:
If you're doing the hauling, plan your route out then check with highways folks (hopefully most of this stuff, btw, can be done with the aid of your trusty computer) for road conditions, construction, closures, etc., that may be planned or happening on that route. After that, check for essential services available along that route (garages, vets, places that could keep horses over night, etc.) and make sure they're still in operation and available. The people I have spoken to have done 'sleep overs' to break their journeys. I remember one who was travelling from Ontario to British Columbia (I'm in Canada and that's a trip that's done in days - not hours) and they did a stop over at a local place by me where the horse stayed at the farm and the people stayed at a near by camp site - they did have trouble getting their horse loaded to resume their trip the next day (I think he had had enough already).
One thing I've gleaned from reading the posts here is to have the paper work in order - brand inspections, coggins test, ownership papers, etc.
Another thing I would do is have date stamped photos in your possession of you standing next to your horses. This provides a bit of evidence that the horses are yours and all so a 'visual' id of them in addition to their paper work.
I'm afraid I can't say much about long haul outfits, if you're going that way, other than they're not all created equal and would therefore require some investigation.
Best of luck with the trip and I see an opportunity for a journey thread (complete with photos).