The 3rd IHG was Cherokee who were initially mounted on their own horses which were "Indian Ponies" (I think technically a pony is an adult horse less than 13 hands tall). Britton mentions the riders' feet almost touched the ground -- undoubtedly an exageration but the appearance would be significantly different than cavalry on horses 15-16 hands.

The ponies were apparently unshod (according to Weer) so their hooves could be damaged by continuous riding on rocky ground. They otherwise had a reputation for being hardier in regards to surviving on grass or doing without food for long periods and being more tolerant of cold weather. However, like Stephen said, by the end of the war there weren't many horses left. Phillips complained often of not having enough horses for scouting or catching Confederates raiding north of the Arkansas.