You would be right to be careful because in total their cultural packages could not have been more different. What they shared a horse and bow military/hunting cultural component.
The Comanche were a hugely powerful people. They controlled a massive territory, levied regular tribute from New Mexico and dominated the trade routes. They were by any contemporary standard wealthy. They made treaties with the waxing and waning imperial powers. They could access as many guns as they wanted. For a while they did so and then reverted to the bow for reasons of military efficiency. You needn’t look further for an explanation. With the advent of revolvers and repeaters they wanted guns again. That was because the new guns were better than bows. They got them too. If you are minded to follow up on this I’d suggest T.R Fehrenbach’s Commanches it’s an excellent read.
I’m sure the Tartars had gunsmiths and powder mills as well as access to those products. They were a powerful entity and close to the Sublime Porte more often than not. For that matter should a Bashkir have really wanted a gun I’d be pretty sure he could get one. Some Bashkirs probably did have guns but Napoleon’s soldiers report them as shooting bows.
All the above leads me to think that it is unlikely all trained horse archers using steppe tactics would totally abandon the bow in favour of the type of gun then available.