When I first started my migration from modern rifles I found a simple recipe for Moose Milk on the 'net, and have used it ever since for patch lube at contests and practice. Also use it to "pre-clean" at the range. Several wet patches for the bore, and a couple damp ones for the lock area. If I will finish cleaning right away when I get home, I will leave the bore as sloppy wet as I can get it. If it may be a few hours, patch dry.
Recipe: 2 oz "water soluble" oil, 2 oz Murphy's Oil Soap, water to make up a quart or liter in a bottle, it's not critical. Add some rubbing alcohol for winter.
At home I remove the lock, soak and brush in warm water and Dawn, dry with a hair dryer and oil. Plug vent or nipple with a toothpick, fill bore with warm water and Dawn, let soak 15-30 minutes, patch really dry, including standing rifle upside down with patch and rod in place for a while.
When barrel is dry, I apply either BC's Barricade (preferred) or WD40. I like to run that patch up and down the bore a few times a couple of days later.
I carry a small bottle of denatured alcohol or can of lighter fluid in my kit to remove the anti-rust oil before shooting again. All this may seem overly obsessive, but the bore on the H & A underhammer bought when I was still in high school still looks pristine.