I take it the barrel is damascus although iron is the same process,the browning you mentioned I have not heard of it ,on this side of the pond I use a damascus brown solution .This is a cold brown and for damascus barrel I do not recommend a hot brown.
First remove all rust if the original browning has either worn off or is badly pitted,take this down to the bare steel .Next wood plug the muzzle leaving enough plug so you can handle the barrels ,next degrease the barrel wash down and dry with a clean cloth,then take a swab with the solution and with long even strokes down the full length of barrel and put aside for 12 hrs.
After 12 hours there should be a good rust on the barrel then holding the barrel by the plug take some medium steel wool and remove the rust,recoat with the solution and let stand for 12 hours ,repeat this operation until the desired colour and pattern is reached this will take depending on the type of steel 5 to 7 days.
After the final de rusting the barrel is dipped in a log wood chip bath.this should be only warm ,it is best the mix the logwood chips with boiling water an let it cool in a dipping tank ,if you dip it in the boiling mixture the barrels will change black.
When the colour has changed to a chestnut colour and enhances the grain of the damascus remove the barrel and dilute the barrel with diluting solution , dry off and warm the barrel with a propane torch and coat with beeswax. This should be left on the barrels for a least a month until the browning has bedded in ,to rub the barrel with a gun oil at this stage will remove the browning.
Damasus twist barrel should using this method should show the twist patten ,on a iron barrel the pattern is of a marble with water marks .
I have enclose images of some barrels I have done ,always remember if the original browning is still good leave well alone only re brown if barrel is pitted ,thick rust,or polished bright.
Feltwad
A badly pitted barrel of a D/B Rifle
The finished job
An etched barrel
A Herring bone pattern from a French sxs Flintlock shotgun
For the locks and furniture these can be toned down with the same solution or use a cold metal brown onlly use one coat do not make too dark.