Wyatt would have been shooting Black Powder. The traditional way to clean up Black Powder fouling has always been with hot water, maybe with a little bit of soap added. And it doesn't take all that much elbow grease to clean up after Black Powder either, the fouling melts right off with some hot water and rags. Trust me on this, I shoot a LOT of Black Powder.
By the 1880s when Wyatt was cleaning up Tombstone, the whaling industry was in serious decline and mineral oil was quickly replacing whale oil as a lubricant.