In the US the key player in the horse flight segment: Tex Sutton’s Air Horse One - Sutton Forwarding
From that link you can even track their flights.
An old article from Sports Illustrated: March 14, 1988 - Horses That Really Can Fly
Each year Sutton transports nearly 3,000 horses in his leased jet. John Henry, Secretariat, Spectacular Bid, Foolish Pleasure, Affirmed and Alydar are all among his former passengers. In fact, of the last 21 Kentucky Derby winners, 17 arrived in the Bluegrass State on a Sutton plane.
Sutton charges about $3,200 for a one-way ticket from New York to California if the plane, which holds 18 horses, is fully booked. The fewer horses there are on board, the higher the fee. Occasionally a good horse will fly cross-country alone—at a cost of about $37,000. Sutton’s business grosses close to $6 million annually.