As said in other answers, revolving rifles aren't very practical and were not widely used. However I wanted to comment on how I think it could most realistically have been done.
Tactically I think such a gun could have been issued to cavalry. Similar to how during The American Civil War, The Spencer Repeating Rifle was not used to replace the standard issue muzzle-loaders Springfield Model 1861/Model 1863 (worth noting the cost of 1 Spencer repeating rifle was $40, which is twice as much as a Springfield Model 1861.)
Side rant for everyone screaming about breech loading rifles: While
breach loaded rifles did very much existduring the 1800s, only 2
countries came close to fully adopting them as their standard service
weapon until after the american civil war: The Norwegian's Kammerlader
of which ~40,000 were ever produced and The Prussian Dreyse needle
guns even thou they only had ~260,000 for 437,262 men by the start of
the austro-prussian war in 1866.
To avoid the problem of cylinder burns, we will use a much shorter barrel than a musket or even a Spencer Repeating Rifle. This means the gun's center of gravity is weighted twords the stock so there's no need to stabilize it with a hand in front of the cylinder. Woah, that means this revolver carbine thing can easily be cocked and fired with 1 hand, unlike the Spencer where you would need to take the stock off your shoulder, press it between the side of your body and your bicep, and then cycle the action. The freehand during shooting presumable means you can ride faster / control the horse better while working the gun, and when you really need the stability you can stabilize the grip by putting your offhand over the hand already on the grip.
For a base I would personally recommend the Colt Army Model 1860 because: it only cost $14.50, permitted easy cylinder removal to allow a quick reload with a spare pre-loaded cylinder - this being an advantage over other revolver designs of the time (1), and was reasonably effective at a distance of 75 - 100 yards (something like a Glock 17 are only effective to ~55 yards).
It turns out that this is actually such a solid idea that Military 1860s had elongated screw lugs on the side of the frame to mount a detachable shoulder stock. (Literally found that out while writing this)
Also as was learned in WW1: revolvers are VERY good in trench runs. Trench Warfare was most prevalent in WW1 but it saw consistent usage even as early as the end of The Napoleonic Wars, especially in sieges like: The Siege of Petersburg and The Siege of Vicksburg. More generally wars like: The Crimean War and NZ Wars saw huge trench systems.
So unlike other answers: not the worst idea. In a military centered around Cavalry (which is pretty good before machine guns became popular) and Cannons/Sieges, I could see most people having a revolver carbine like weapon.
Another rant about why using a sealed cylinder with rifle cartridges
wouldn't work: The seal would need to be significantly 'heftier' to be
usable with rifle cartridges. Then the cylinder's diameter would need
to also be significantly increased to fit the added heft between the
cylinder axle and the bottom of the seal. Also there is spring between
the front of the frame and cylinder, around the seal / tube, which
pulls the seal and cylinder shut. This spring would need to be much
stronger so that the cylinder doesn't open when firing a higher power
(1) This was probably not widely used by the US Army as none of the guns were issued with a spare cylinder, but I'm guessing it's feasibly as an extra cylinder is probably at most $5 more