Formby's Tung Oil in Satin sheen works incredibly well. Very durable and self levels since it has a low viscosity. Degrease barrel, I use acetone. Then I cinch a dowel rod horizontally in my vice that is large enough to go into the chamber but jams in the forcing cone. Slide the barrels onto the dowel and proceed. I wet a patch made from a blue shop towel with the tung oil—it has virtually no lint. Apply it fairly liberally but not so much as to run. Run the patch the entire length of the barrel, turning it and repeating until entire surface is covered. It will first appear to be too much but the Formby's has so little solids in it that when dry will be a very thin coating. It will self level. I also use the same for actions and other metal parts. Don't worry about filling screw slots or engraving--you won't. This stuff really holds up well and I have not had any flaking or peeling. Oh, it dries quickly, usually within one day and can be handled. Fully cures in a weeek. The finish really gives the damascus finish a depth that really brings out the color.
Now, I should mention that if the barrel is heavily etched to the point its almost rough you'll be better off not applying any finish at all, maybe some wax. The sheen on such a barrel is distracting in my opinion.
