One of the primary advantages of a muzzle loader where the barrel is retained by a hooked breech plug on one end and 1 or 2 wedges along the barrel is that the barrel can be quickly and easily removed from the stock and separated from the lock by removing the ram rod and the wedges.
This feature lets you remove the barrel for a thorough cleaning without getting water in the stock or lock.
The muzzle loader can be cleaned very effectively in this manner:
1) remove the barrel from the stock;
2) unscrew and remove the nipple and let it soak in a cup of hot soapy water;
3) fill a gallon bucket or similar container with hot soapy water (tap water as hot as it will get);
4) slowly run a patch on a suitable sized cleaning jag up and down the bore several times, which pumps the hot soapy water in and out of the bore through the hole for the nipple (if you pull or push too fast, you'll push excess water past the patch and make a large mess as the now dirty water is pushed out of the bore on the wrong side of the jag, you'll get some anyway, which is good as you want to ensure the bore is clean at the muzzle as well);
5) with the bore wet, use a bronze bore brush to loosen any harder fouling, and repeat the flush with hot soapy water several more times;
6) wipe down the outside of the barrel with hot soapy water and use a tooth brush or similar to brush the area around the breech end of the barrel to ensure any powder fouling on the exterior is removed, and neutralized;
7) refill the bucket with clean water, again as hot as you can get it, and use a clean patch on a jag to pump water in and out of the bore several times;
8) by this time the barrel as a whole should be hot enough that you' can't hold on to it for longer than a few seconds. Run a few dry patches down the bore to remove the water, and then let it dry. The latent heat in the barrel will help any remaining water evaporate.
While the barrel is drying clean the nipple with a toothbrush and a pick and then rinse it or soak it in hot tap water, then dry it and set it aside to let the heat do the rest. Then wipe/brush down the lock and breech plug with a black powder solvent to ensure any corrosive residue has been neutralized. Dry and re-oil the surfaces - including the inside and outside of the nipple. It's not necessary to remove the lock and breech plug every time, but it's a good idea to do so a couple times a year, and before putting the rifle up for storage over the winter.
Once the barrel has dried, clean it like you normally would any smokeless rifle barrel. Use a powder solvent, first on a wet patch, then with a bronze bore brush to help remove any lead deposits, and then with wet and dry patches, to displace and remove any residual moisture and fouling that may have leached out of the barrel. Then run an oiled patch down the bore. Wipe down the out side of the barrel to replace the oil that will have been removed by the hot soapy water.
Since the corrosive salts in the powder residue can be driven into the barrel steel and will leach out over a few days, it's always a good idea to run an oiled patch down the bore a day or two after you've cleaned the bore to remove these salts and ensure the surface of the bore is well oiled to help prevent rust if any salts remain.
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The next time you are ready to shoot, run a dry patch down the bore to remove the excess oil, and then fire a couple of caps before loading the rifle. This will ensure the nipple is clear and will ensure reliable ignition.
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Alternatively you can skip the hot soapy water and just use a black powder solvent to clean the bore, but it takes longer and isn't any more effective. You'll still want to follow up with a regular solvent and oil as those black powder solvents are water based, since it is water that does the neutralizing of the corrosive compounds.
Some folks recommend Windex as a bore cleaner, on the basis that the ammonia in the Windex neutralizes the corrosive salts. That may well be true, but the water in the Windex is just as effective, and you might as well just use hot soapy water and save the Windex for the windows.
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If you shoot corrosive primed ammo in your smokeless powder rifles, you'll find those same black powder solvents work great to remove the corrosive priming compounds. I used them on the bores and in the gas systems of my AKM, SKS, and PSL and they do a great job of preventing rust. Again, dry and oil the surfaces after you've cleaned them with the black powder solvent.