Isaiah Jennings patented an improvement to the Belton repeating flintlock system in 1821 - but we don’t know exactly what his idea was because the Patent Office lost his patent (and many others) in a large 1836 fire. Jennings' system was used by several gunsmiths, though. In 1828/9 the State of New York contracted to convert 521 of their muskets to Jenning’s-pattern repeaters. We also have a few examples like this custom 5-shot pistol made by John Caswell of upstate New York.

Jennings' system uses superposed charges loaded in the barrel along with a movable lock. Each charge has its own touch hole, and the cover plates for them act as stops for movement of the lock, to ensure proper alignment. The trigger will fire the lock in any position, and it is also fitted with an automatic magazine frizzen - so cocking the hammer automatically charges priming powder into the pan and closes the frizzen. These were very advanced arms for the early 1800s, and expensive to produce.

Belton Repeating Flintlock:
https://forgottenweapons.vhx.tv/videos/belton-flintlock-4k