Horses are not native to North America but they were used by Native Americans at a certain point. How long did this process take from introduction to incorporation? How long did the knowledge of horses take to spread throughout the American continents?
First thing: Horses actually are native to North America. They died out during the last ice age; however, groups of them had traveled across the Bering Land Bridge into the Asian Steppes, where they flourished, and were eventually domesticated by the ancestors of the Mongols.
Spanish explorers first brought horses to the New World around 1520, but that was just a handful brought by Cortes. Coronado brought more than 500 horses with him when he invaded in the 1540's. By the early 1600's Spanish settlers had spread up into New Mexico, founding the town of Santa Fe. They brought entire herds with them. They made it illegal for the local Indians to own horses or guns, but many Indians employed or enslaved by them did learn how to work with horses, how to ride them and how to care for them.
In 1680 the pueblos in northern New Mexico revolted, successfully driving the Spanish out of the area for a dozen years. The Spanish fled hastily, leaving many of their horses behind, which the Indians took control of. They began to breed the horses, and to sell or trade them to other tribes such as the Kiowa and the Comanche. The knowledge and ownership of horses spread quickly. By 1700 the tribes of the Great Basin had them, and thirty years later those of the northern Rocky Mountains. By 1780 they had spread all the way to the Columbia River Basin in Washington state.
A couple of good references that have further information: