The origin of the American Indian horses are horses brought over by “Spanish intruders” from Europe to the New World. Columbus took 30 horses to America in 1492 and Cortés took 16, and arrived in February 1519. They gave rise to “a whole new population of Indian horses”. They changed “the social structure of the Plains Indians of the Americas”.
The breed of horses that they took with them would have been the Spanish Barb and Arabian, as I understand it. The Mustang of the Western United States is descended from these horses brought to the Americas by the Spanish. They are described as wild horses but in truth they are feral domesticated horses. The Barb is a North African breed of riding horse. There were imported into Europe from the early eighth century onwards. The Barb horse was crossbred with Spanish stock. It is said that the influence of the Barb is seen in the American Quarter Horse, the Mustang and the Appaloosa.
The Arabian is one of the oldest breeds dating back to about four and half thousand years ago. They were transported around the world by traders and used to improve other breeds in terms of endurance, refinement, speed and bone strength. They are found worldwide including in the United States.
The source of the information on this page is Dr Desmond Morris in his book Horsewatching and Wikipedia. The quoted text is from Horsewatching.