A cavalryman would typically carry musket-size caps for his carbine / musketoon / rifle in his cap box.
The pistol- / revolver-size caps are so small that it would be impractical to fit them on the nipples while in the heat of battle, so to speak.
One would be better served by swopping pre-loaded & capped cylinders.
Combustible cartridges were the preferred ammo, but especially the Confederates would frequently have to rely on cast roundballs and a powder flask.