With a blued barrel, it is critical to get the water out of it after cleaning. The worst flash rusting problems I had was when I was cleaning with hot water back in the 90's, which I no longer do.

I use only a BP solvent, brush and patchs (Butch's or TC #13), followed by a water displacer (Birchwood Casery, WD-40, Remoil) and wipe until dry. Then store in a low humidity cabinet or safe (20% RH or less). I do not use oil, except sparingly (a drop or two) on locks.