The only thing I know for sure is that you need to provide LOTS of
water, and drink more yourself (gallons!) since the biggest problem with
going to a higher altitude is dehydration. You could come to the higher
altitude and sit around doing nothing and loose moisture. We see lots
of tourists, even fit ones, usually robust energetic young males, who
are altitude sick (headaches, nausea, short of breath) and they've been
drinking "lots of coke, coffee, tea and alcohol" - all dehydraters.
Other than that, I guess the horse, and you, might need a few days (a
week?) to adjust to less oxygen. The only cure after the dehydration
damage is done is to go to a lower altitude.
Bambi (live at 7,400 ft, and horses still breathe harder at 10,000, but
they love to go to the Moab, UT area!)