It depends really on how far you went with the browning. My opinion anyway. I nearly always when the browning is done, wipe down very good with 4 ought steel wool and a good oil. Then in a few hours, or maybe the next day, there will be some rust appear. I scrub that down again, and then begin to scrub with RIG and four ought steel wool The edges of the flats might begin to show but they nearly always settle down afterwards.
I then wipe with the rig and a good rag. I always considered this part of the process. It gives a lot of depth to the brown and darkens it a lot. Then just regular MTn. and oil every once in a while as you would anyway.