The transition of traveling by foot to traveling by horse occurred by at least the late 1600s and perhaps much earlier. The Southeastern Tribes probably changed from "no horses" to "everyone has a horse" in a generation just as they did with firearms. At the beginning of the war, common or lower economic classes had lots of these "ponies" and virtually every family had one or more. The middle and upper economic classes had Thoroughbreds and other breeds. Horse racing (and gambling) were very popular in the Indian Territory, especially around the military posts. It is my impression, like Stephen suggested, that the Tribes of the area were quite adverse to being an unmounted force.