Usually they "express" the semen and add it to an egg extracted from a promising mare into a Petri dish, then
implant into a brood mare. No one (at least around here)
actually allows mating the oldfashioned way - too dangerous for the stud (MARES KICK)!
Thoroughbreds are only bred the 'old fashioned way' (there is no AI allowed) and FWIW, so are many, if most horses of other breeds bred the natural way even where AI is allowed. If the stallion is reasonably local, it's better to do it in person, and, rather than extracting any eggs, the insemination actually is more typically done IN the mare while in season.