The horses I've seen brought into our area, just haven't had a hard time with the altitude. If they are in shape, they do OK. If they are pasture potato's then they have a problem.
I live at 4472 foot elevation and frequently ride over 11,000 foot summits in the summer and fall. Yes my horse need to stop and blow. But they catch their wind and are ready to move on in a few minutes. I put on a CTR/Endurance race up at 8000 foot with some of the trails crossing 11,000. We had horses come in from Georgia, the Carolinas, California and they all did fine.
I see more problems with the riders. People drink diuretics that dehydrate them. Coffee, caffinated soft drinks etc. If they stayed hydrated with lots of water they probably wouldn't have gotten sick.
I've seen ranchers come up deer hunting. They spot a deer and basically chase it until they get with gun range. Their horses are faster than the deer, even going up and down steep mountains. I suspect that comes from chasing cows on the mounain all summer. My horses will make it up and over the mountain with a couple of stops. But they won't keep up with the deer.
So basically, if you watch your horse and give them the occassion blow, They usually do fine in the higher elevations.