I've taken a percussion rifle on a late season hunt into the backcountry a few times and have gotten some great tips from others on this forum. Where I go, it's VERY humid at camp in the creek bottoms but often pretty dry where the deer are up on the hillsides. Sort of depends on the weather, though. I'm also not likely to take more than 1 or two shots over a 4-5 day hunt and, once I shoot, I'm coming back home within 24 hours -- not staying out. Here's a few things I do or carry for that hunt.
Carry 2 small squeeze bottles -- one with dish soap and one with alcohol. I mostly use the soap for poison oak though. The alcohol would be used to clean if I happened to miss an early shot and spend a few extra days down there.
Use rain protection -- the small condom-like cover on the muzzle and the pink surgical tubing thing over the nipple. I also coat the nipple threads and cleanout screw (traditions rifle) with beeswax/bear grease (or oil) mix.
Bring a .22 wire brush for cleaning out the breach.
Nipple wrench for cleaning.
A dozen pre-greased patches, maybe 2 doz dry. I also have a strip of old t-shirt tied to the outside of my possibles bag to wipe stuff when needed. So far, I've only used it to scrub poison oak off my skin with some hand sanitizer if I happened to touch it during the day. But I can cut strips for more cleaning patches.
Usual accoutrements (ball starter, patch worm, jag, etc.).
Ball puller -- kind of optional. If I know it's going to be wet, I'll take it. Otherwise, I'd just dribble powder into the drum to get a stuck ball out.
Pre-measured powder charges in vials. Again, I only take a handful since I'm not doing much shooting. A powder flask & light weight measure would be better if you plan to do some roaming or small game hunting.
Small multi-tool -- mostly for backpacking. But I have yet to use it more than once even though I've been carrying it for 15 years.
Some kind of screwdriver or tool to take the rifle down for a pretty thorough cleaning. I can do most things with just a homemade flathead screwdriver tool with a pick on the end and a nipple wrench.
Pipe cleaner
Extra nipple
...
I'll try to think of some other things. Change as needed for a flintlock.