Certainly the majority of Union cavalry troopers at least had breech-loading single-shot carbines. A few were armed with repeaters like the 7-shot Spencer. I'm not sure if loading on horseback was all that typical, frankly? I think most cavalry used the horse for mobility, and every now and then for charges or hot pursuit of the enemy, but often fought dismounted like Dragoons.
In the CSA, cavalry sometimes had their horses taken away/ requisitioned, and the units fought as infantry.
Loading an Enfield P53 rifle musket with the U.S. type cartridge would be a real chore while mounted. On foot, it would be no different from what the infantry did.
I do know that the former 1st Mississippi (U.S.) cavalry, later the 3rd U.S. Colored Cavalry used the Model 1855 Colt Root revolving rifle, which was mighty unpopular with Berdan's Sharps Shooters (who wanted Sharps, after all), but was the equivalent of what lots of Iowa cavalry carried on Grierson's Raid, and was used to telling effect by the 21st OVI at Chickamauga.
There have been past threads on the armament of the USCT that might be of interest, or germane to your topic, although I think these were mostly on the "3rd class" and "2nd class" weapons for infantry. I think that by the later parts of the war most USCT carried comparable weapons to any other Federal outfit.
https://history.army.mil/html/books/030/30-24/cmh_pub_30-24.pdf