A humidity box is another option for finishing the barrel. Make a box long enough to fit the barrel, have a pan of water in it and a 100 watt light bulb. That will do the thing.
Personally, I built a "cabinet" 2'x2', installed a 100W light bulb in the top, a hot plate in the bottom (set at a little less than medium), set a metal coffee can full of water on it, closed the door and didn't open it 'till the next morning.
I would normally repeat this twice, maybe three times, card the rust, then use Baking Soda to stop the process.
It worked pretty well, however I have seen other "browns" I like better than mine.
Browning is much like Bluing...the more preparation you put into it, the better the end product.
I have seen some "Plum Browns" you could shave in, and other browns that looked more dirt.
My own personal preference was somewhere between, and that is what a "Rust Box" will give ya..
I'll post a picture of my best effort to make a "Rust Box".....the barrel in the box is on it's third trip through, notice the front sight installed on the barrel....if you don't take 'em off and get a good even brown "under" the sight, it's going to show later on.
So the last trip through the process you re-install the sight.
Uncle Russ...