I would suggest asking potential shippers about their policies on hauling horses long distances.
Some will stop every so many hours. One we used was on the road approximately 20-24 hours, then they stopped at a layover barn, unloaded the horses for the night, next morning they loaded up & continued on--our horse only had 2 hours on the trailer after they resumed their trip. Another hauler that we have used stops every night--she drives 12-14 hours and then stops for the night; the horses get unloaded at a layover barn. These shippers use the same layover barns every trip, so they know the facility and owners can be assured that their horses are being well looked after at these barns.
Find out if the shipper makes the trip as direct as possible, or if they will pick up your horse and then zigzag all over the country picking up/dropping off horses--your horse might be moving only 500 miles away from home, but he'll ride 2000 miles on the trailer while the rig travels all over the map. Choose a shipper that is making the trip following a specific route.
Some haulers do offer box stalls, which would afford your horse the opportunity to lie down while in transit. (Some horses will lie down while on the moving trailer, others won't) Be sure to ask what kind of stall is available for a Mini on the shipper's trailer--you may not want him to be put into a common area with other horses, particularly if they happen to be full size horses (some haulers do have just 2 or 3 compartments on their trailer, and the animals all ride together in a group in each compartment. Others offer individual stalls.) Find out if the dividers are set up for minis, or if they are full size horse dividers, which may not be suitable for minis. (too high--mini can fit under the divider).
If you're concerned about how your horse will be shipped, it pays to ask lots of questions, don't just assume that haulers are all the same and the one that is selected will haul horses the way you assume they will be hauled.