They used mounted archery, but not always with the same kind of tactics as used by Steppe armies. There appear to be two major reasons for the differences:
Battles usually involved far fewer participants than large battles involving Steppe armies.
The gun was acquired at about the same time as the horse.
In addition, there was no established system of military discipline with soldiers taking orders from officers.
Steppe tactics that were absent from the Plains included mounted archers supporting armoured cavalry (mounted cavalry saw limited use on the Plains, but disappeared as guns became more common), the use of armoured mounted archers, and large formations of mounted archers. Steppe archers would sometimes use archery at very long ranges, where their chances of hitting were low, to harass enemy forces, while Plains archers would usually reserve archery for short ranges where the chance of hitting was much greater (Steppe archers would also shoot at short range).
Since the guns in use on the Plains outranged bows, but until the late 19th century had lower rates of fire, mounted archers would often use loose weaving formations to attempt to avoid gunfire while closing to effective archery range, or dismount to take advantage of terrain for cover to approach enemy forces well-equipped with guns.
Tactics and techniques such as feigned retreats, groups of mounted archers dispersing to avoid enemy cavalry charges, and using the horse for cover while shooting were common to both Steppe and Plains.
References:
R. Bohr, Gifts from the Thunder Beings: Indigenous Archery and European Firearms in the Northern Plains and Central Subarctic, 1670-1870, University of Nebraska Press, 2014 (notably, chapter 8).
T. R. Fehrenbach, Comanches: The History of a People, Alfred A. Knopf, 1974