Fed Ex, UPS and large commercial airlines ship horses and other animals as cargo, but the Tex Sutton company and the Kalitta Charters Co use a dedicated aircraft to do so.

Ferrying Kentucky Derby winners and other prized horses began by air in 1969, now moves horses for the Olympics, Belmont Stakes, dressage championships, etc

Rodeo horses never fly because the professional circuit schedules its events so that horses can be transported economically by vans the relatively short distance to the next stop on the tour. And, perhaps surprisingly, horses used in polo – a sport supported almost exclusively by the mega-wealthy – rarely fly.

Horses are loaded into specially built stalls that can be arranged two or three across inside the airplane. While the owners of some “celebrity” horses may charter the entire plane, Air Horse One can carry 18 to 20 horses per flight.

To accommodate their special cargo, the pilots of Air Horse One make wide turns and extra-gentle ascents and descents to try to keep the horses from getting spooked or losing their balance.

"Air Horse One," is operated by Kalitta Charters II. 

Doug Kalitta, owner of the charter company, is the nephew of Connie Kalitta, who is equally famous in the air cargo world for his pioneering use of aircraft – from little Cessnas to giant Boeing 747s in specialized chartered air cargo operations.

https://www.usatoday.com/story/travel/flights/2016/10/26/tex-sutton-air-horse-one/92782198/

https://www.forbes.com/sites/danielreed/2016/06/09/belmont-air-horse-one/