Real talk they served as light cav units which were recon + raiding supply areas like fresh water or caravans, plus constant skirmishing with enemy armies before battles would be joined.
Typically in a pitched battle both army's cavalry will engage each other first on the flanks, more often then not the side who's cav survived intact will go on to horse bump/couch/headshot/troll the enemy infantry which would often lead to an instant collapse in morale and their routing.
Bow cav would be part of these engagements and would typically engage each other, keep in mind in cav vs cav fights even thrower or bow cav would often just start punching and grappling the shit out of each other, get dismounted, then stab each other to death, things would be insanely hectic and this was often the most important part of a battle (which cav players had more balls). After enemy cav is dealt with, bow cav just surround the enemy formation and shoot from every direction.
Mongols were famous for using feigned retreats and other deceptive, mobile tactics with their armies that were primarily bow cav. Armies that were all mounted like this primarily had lancer and bow cav working in tandem, like the Battle of Carrhae someone linked.
Basically bow cav would often engage enemy cav who would pursue, and bow cav would retreat while shooting over their backs. Then lance cav would outflank the pursuing cavalry, and wreck them.