The rust you find in bores normally is " Flash Rust", that comes about from using TOO HOT water when cleaning, and then forgetting to put an oiled patch down the barrel quickly enough. Or leaving the gun sit in a damp location, where the oils dry out, and the metal oxidizes.
Use a bore brush- they are bronze, not brass-- and you can find brushes that are made with synthetic bristles, but not for all ML bore sizes---- with some fine (OOOO) steel wool on it. The steel wool has an oil in it, but use the steel wool to simply remove the surface rust from the bore.
Follow it with an alcohol flush. Pour some alcohol down the barrel, after plugging the vent hole, or nipple hole, and then putting your thumb over the muzzle. Shake the barrel back and forth, like you are making a mixed drink at the bar. Then, pour the alcohol and crud out of the barrel. Then run a few patches down to get the rest of the crud, and help the alcohol dry faster.
Now-- and this is the most important step-- use an oiled patch to lubricate the now-bare metal of the bore. WD40, Ballistol, olive oil, Jojoba oil, and any number of oils extracted( rendered) from the fat of animals, like deer, bear, groundhog, raccoon, etc. work as lubricants, that will also displace water from the surface of the metal.
Of course, as the alcohol in the barrel evaporates, it will take with it any remaining drops of water. Your choice of oils then for lubricating the barrel for storage, or for field use can take in other concerns than water displacement.
Testing done here shows that Ballistol works well to protect steel over long storage periods, but the best product mentioned so far is the Birchwood Casey " SHEATH". There are also some synthetic oils out there that do very well in extreme Climate conditions.
The main idea to remember about any storage oils or grease is that THEY HAVE TO BE REMOVED FROM THE GUN BEFORE TAKING IT OUT TO SHOOT AGAIN.
Use alcohol as a solvent to clean(flush) the stuff out of the barrel, flash channel, vent, or nipple. Then, put just a very light coat of oil in the bore to protect it from rusting during transportation from you home to the range/field.
At the Range/field, remove the oil with a dry cleaning patch or two. Either use a pipe cleaner to clean the oil out of your flash channel, or consider firing 4-5 caps off on you percussion action, and then wiping the bore with a cleaning patch to clean out that residue, before putting a powder charge down the barrel. If you shoot a percussion gun with a "Powder chamber" you will need a reduced size cleaning jag that allows you to get down into that chamber and wipe it clean, too.
These last steps are what most shooters DON'T DO, and is what causes them problems shooting black powder at the range, or in the field.
If its going to be a long sit in a treestand when hunting, you might consider running an oiled cleaning patch down the barrel AFTER YOU SEAT your PRB, or conical on the powder. This protects the bore from rusting in the field. I use Wonderlube, just because it "sticks" better, with the beeswax in it, but any kind of oil is better than none at all.
If its raining or foggy, or misty when you go out into the woods, its particularly important to protect the metal parts by wiping them with a greasey( waxy) cleaning patch- not enough to be sticky, but enough to give the metal that waxy feeling, and to bead water dropped on the metal. If water beads on the outside of the barrel as the result of whatever you wipe the barrel with, you can be assure that it will also not reach the metal inside the bore, provided you wipe the bore with the same stuff, too.
