I've been shooting flinters for a few years now and this is what I do.
1) Get a wooden round toothpick.
2) Fill a glass with hot, hot water from the tap.
3) Dip the toothpick in the hot, hot water for a minute or two and then stick the toothpick in the touch hole.
4) Pour enough hot water down the bore to mostly fill it. Let the water sit in the bore for a few minutes.
5) Cover the muzzle of the flinter with your finger and slosh the water back and forth. Pour out somewhere where blackish, icky, water will not do any damage (ie. not the lawn or the garden. Probably a bucket which can be emptied down the sewer)
6) With the rest of the water in the glass, repeat steps 4 and 5. No need to wait a few minutes this time. Also, remove the tooth pick and let about half the water drain out of the touch hole.
7) Dry the bore as well as you can with dry patches. It should take about three patches; no more than four. (Please note: you will never get a perfectly clean, pristine patch - don't worry about it).
8) Oil a patch with the oil of your choice. Run it up and down the bore. I use olive oil because the gentleman who made my custom flinter recommended it. His whole life is using original materials. Today's olive oil is the 18th century's sweet oil.
9) Repeat step 8 at least two more times.
10) Oil the lock and the touch hole.
11) Clean the stock by whatever method suits you. I used to use olive oil on the stock as well but after a year or two or three of this, a somewhat sticky layer begins to build up. I am now in the process of removing the sticky layer, with pretty good success.
12) Admire your flinter and put it away until next time
With a little practice, you should be able to do all the above in a half hour or less. No need to remove the barrel.
Works for me.
