The First Horses in Mexico

The sixteen horses for Cortez's expedition were purchased in the Caribbean.

Here also we hired soldiers, and purchased some horses, which latter, at that time, were very scarce, and only to be had at exorbitant prices. Alonso Hernandez de Puertocarrero, whom I had previously known, had not sufficient money to purchase himself a horse; Cortes, therefore, bought one for him, and paid for it with the golden borders of the velvet robe he had procured at Santiago. (Diaz, 1568)

Diaz (1568) goes in to great detail about the specfic traits of each mount.  It is noteworthy that the mounts included both mares and stallions and that all horses were said to be trained in both jousting and war.  

For memory's sake I will here likewise describe the horses and mares which we took with us on our expedition. Cortes had a dark chestnut stallion, which died afterwards at St. Juan de Ulua. Pedro de Alvarado and Hernando Lopez d'Avila had jointly an excellent brown mare, which had been broken-in for the field of battle as well as for tournaments... Alonso Hernandez Puertocarrero had a grey-coloured mare, which Cortes had purchased for him with the golden borders of his state-robe, it was capitally trained for the field of battle. Juan Velasquez de Leon's mare was of the same colour, a noble and powerful animal, full of fire and eager for battle: we commonly termed it the "short tail."  (Diaz, 1568)

Christobal de Oli had a dark brown fine-spirited horse. Francisco de Montejo and Alonso de Avila had between them a sorel-coloured horse, but of little use in battle. Francisco de Morla had likewise a dark chesnut stallion, one full of fire and wonderfully swift. The light-coloured horse of Juan de Escalante was not worth much. The grey-coloured mare of Diego de Ordas, which would never foal, was neither very swift. Gonzalo Dominiguez had a small dark-brown nag, a very swift and noble animal. Also the brown-coloured horse of Pedro Gonzalez de Truxillo was a swift animal. Moron, who was a native of Vaimo, had a small horse which was pretty well trained. Vaena, of Trinidad, had a darkish-coloured horse, though a bad leaper. The light-coloured chesnut galloway of De Lares was, on the other hand, a splendid animal and a capital runner.  (Diaz, 1568)

Ortiz, the musician, and a certain Bartolome Garcia, who had applied himself to the art of mining, had between them a very good dark-coloured horse, which they named the Arriero (mule-driver,) and was one of the best animals of the whole corps. Juan Sedeño, of the Havannah, had a fine chesnut mare, which foaled on board. This Sedeño was considered to be the most wealthy man amongst us; for he had a ship of his own, a horse, a few negroes to attend upon him, and his own lading of cassave and cured bacon. Just about this time horses and negroes were only to be purchased for very high prices, which accounts for the small number of the former we had with us on this expedition. (Diaz, 1568)