Not many stallions are shuttling, and those who do only usually a change of ownership which means an occasional venue change.
It’s the mares who have to shuttle.
Of course mares go to the stallion in the region. I was referring to those that shuttle to Australia or South America of which several notable stallions have, including American Pharoah. It can be tough to cover close to 200 mares (up to 3 a day) in a space of 3 months and then travel a distance to stand a season in a foreign environment.