Howdy
Well, here's what I do. I take an empty case and chamber it and close the lever. Then I stand the rifle up vertically. Then I take a patch, insert it in the slotted end of my cleaning rod, soak it with my favorite water based BP cleaning solution, and twirl it down the bore. Do not use a jag, use the slotted end of the cleaning rod. Pull it up and it will be covered with black crusty fouling. Repeat. After each patch, they come out cleaner and cleaner. When the patch comes out dirty gray, with no crusty stuff, all the fouling has been washed down into the empty case in the chamber. The bore is basically clean at this point.
Turn the rifle upside down and eject the empty case onto the ground. A spray of filthy, black powder solvent will come out with it, so don't do this on the wife's new white carpet.
I shoot nothing but 44-40 in my lever guns, which seals the chamber beautifully, and almost no fouling gets past the case into the chamber. You will probably have a bit more with a 357 Mag or 38 SP case. Anyway, take another patch soaked with solvent and clean around the bolt, the carrier, and anywhere in the action where fouling may have accumulated. Follow up with some straight Balliistol down the bore and on the carrier and bolt. I like to soak a patch in Ballistol, twirl it down the bore, then follow up with a dry patch to mop up most of the Ballistol and just leave a light coating behind.
This entire procedure only takes about ten minutes. I DO NOT take the side plates off my rifles and clean out the insides every time. I have previously coated the insides of every thing with Ballistol. Ballistol (or any oil) will soak down any fouling that gets into the action and prevent it from wicking moisture out of the air. BP fouling that has been saturated with oil is harmless and will not cause rust. I only take my BP guns apart maybe once a year, to clean out all the oily, black guck down inside. There is always plenty of oily black, guck there is never any rust.
P.S. The reason you don't use a jag is it can get jammed in the empty in the chamber. The slotted end of the rod is narrower and less likely to jam.