Caroll, a non-player character, owns all HR Foundation facilities of the game world. She offers a selection of horses from each breed, except the wild breeds. There are always 4 horses of a breed available for purchase. To buy one, click on the purchase button under the horse on the overview page. Foundation horses cost 5000 HRC and 1 Foundation Ticket (FT) each and are first come, first serve. If one of them gets bought, a new foal of the same breed will immediately replace it.
All foundation foals spawn at 6 months old. They age regularly, regardless of whether they get bought, and will stay at the HR Foundation into adulthood unless bought. The only time foundation horses get replaced is when they get purchased by a player.
On each continent, there is a set of 10 mares and 10 stallions per breed which constitute the local foundation parents. These are not up for sale or studding. They get paired at random to produce foundation foals for that breed on that continent. Traveling can pay off: Each continent has its own foundation stock and, as a consequence, different bloodlines. Accessing the HR Foundation will always show you the available horses for your current location. If you buy a horse from abroad, a transportation fee of 500 HRC gets added to the price, totaling in 1 FT + 5500 HRC. The horse will automatically appear in your stables at home.
You can't directly see the stats of foundation horses prior to purchase. There is only the Breeder's Advice on the genetics tab of a horse. This report will give you a rough idea of what to expect, but the actual stats of foundation horses only become visible after you buy them.