The Henry would have used black powder and needed many cleanings during firing, at least cleaned once every 10 - 15 shots. They probably had cleaning bullets to scrape out the fouling in the barrel, which would have been considerable, and caused dangerous pressures because of reduced barrel diameter.
However seems like the action itself would be gunked up as well after many firings. I've never heard that the Henry was bad about jamming, which testifies to its fine craftsmanship.
It did have one flaw I believe, in that the magazine tube beneath the barrel had a viewing slit, so the shooter could see how much ammo was left. I may be mistaken about this, but if true, it would allow dirt into the magazine.
I've read that due to the nature of the Henry's construction, with its (brass?) receiver and great usage of metal parts throughout, it got very hot to handle after a few shots. Some of those Henry shooters in combat would keep a bucket of water handy to dip the barrel into to cool it off a bit.
The centerfire falling block Sharps appeared in 1873 as an Army weapon and was in use for 6 months, when it was replaced by the Winchester '73. Bear in mind smokeless powder was not to be invented 'til the 1880s, so even the '73 would have used black powder.
Lots of smoke and heat and probability of ammunition "cooking off."
Les