I have used HOPPE'S No. 9 lead solvent, an old, reliable modern gun solvent, to remove the little bit of lead that gets in the barrel. I have also used Shooter's choice, which works very well. A good bore brush takes much of the work out of cleaning lead out of a smoothbore. I like to dop the brush into the solvent, and then run it down the barrel in a ' back and forth " actuin to scrube the barrel while the solvent goes to work getting under the lead streaks, and allowing the brush to break them up and off the steel bore.
I found early on that if I first wipe out powder residue( crud ) with a damp patch or 3, and then use lead solvent on a cleaning patch that is fairly well soaked with the stuff, and put that down the barrel leaving the RR in the barrel for the trip home, By the time I get home, the lead is loose, and comes out fairly easily on the cleaning patch with the solvent that took the ride home in the gunbarrel. With my DB shotgun, I can't obviously have two rods down the barrels, but I found that a good soaking of one barrel, and then another good soak and wipe of the second, leaving that cleaning patch with solvent on the jag, and the RR down the barrel allowed me to clean up both barrels very quickly when I got home. After a particularly long shoot, I had to soak the first barrel longer, so I just put more solvent on a new patch, and ran it down the barrel, and left it sit while i put the rest of my stuff away. Then I cleaned the lead out of the barrel in long streaks, flushed, it, dried it, oiled it, and put it away. I do put the gun muzzle DOWN on newspaper, and paper towel for the first few days, so that any excess oil comes out of the barrel onto the paper toweling. Then I case the guns and leave them in my gun closet. I always check all the screws on my gun to make sure they have not worked loose. If I have a screw that seems to make that a habit, I dig out my LockTite, and put a drop on the threads to end that. I always check the gun's action to make sure its functioning properly, is cleaned and oiled, and ready to go. That way, I don't have to spend a lot of time preparing my guns before I go out shooting the next time. If I have not had the gun out in several months, I clean the gun and re-oil it, and check all the working parts and screws the night before the gun goes out. The process allows me to get familiar with the feel of the gun, and give me the confidence that the gun is ready to use, and will function properly. I used to use WD40 as a lube, but after I had a revolver malfunction because the stuff had dried out and gummed up the works, I cleaned all my guns of the stuff, and have changed to using RemOil, a commercial gun oil. I am sure there are many good oils you can choose to use. I just happened to find this on sale when I went looking.