A few thoughts.....
I always use a flat-face brass scraper on the end of my rod first before anything wet goes in the bore of a flat breech flintlock. It scraps the hard gradeau loose from the breechface. I turn the rifle muzzle down and tap lightly to get the crud to drop out. Why get it out the hard way?
A flush kit is a great thing, and simplifies cleanout. TOW sells a good flush kit. They have them for flinters and cap shooters.
I use water to clean out my barrel initially.
If I've forgotten my flush kit, I plug the touch-hole with a toothpick and trap a patch with the pick under the frizzen. The patch catches any minor drool. I then pour water down the barrel and let it sit a while, then slosh it back and forth. I then repeat it. You get lots of fouling out that way. I then go to wet patches until they come out clean. Sometimes quickly, sometimes not.
Often you'll see black streaks on the patch. Look to see if the streaks are where the jag head is, or behind it. If behind the head, your rod may be dirty and transferring to the patch. I drove myself crazy one day until I figured that out. Cure? Wipe the rod with a wet patch as you get close to clean.
If the fouling is super bad, I'll use a nylon bore brush. I do NOT use a brass one. Brass ones can get stuck at the breech end. I also don't want anything scratching my bore to make it harder to clean.
After I am done, I dry the bore well and use my favorite bore protecting oil.
If you have a patent breech or any chambered breech, store your rifle muzzle down. The bore oil can flow into the breech and gum up. Taylor and Daryl had to save me on one I bought. Great advice guys, and thank you.
I agree 100% with Daryl on checking your gun a day or two after you clean it out. Run a few more wet patches down. You'll often see a little light film rust. I have been told that you never get all of the moisture out the day of shooting, so let it evaporate and re-oil. No problems after that ever.
I ALWAYS clean my guns completely at the range. No excuses.
Last thought: some days the fouling will harden up on there like charcoal mixed with epoxy. You'll spend a long time getting them clean on that day. I had it happen last week on a really dry day. Shoulder still sore from that cleanup.
Best wishes and God Bless, Marc