There is an old saying,,,"If it moves and it's not supposed to, use Duct Tape, and if it's supposed to move and doesn't, use WD-40."...Sadly, there was a generation, much after mine that thought WD-40 was a cure-all for everything...Mine was 3-in-1 Oil...
As stated, you will probably need to totally remove the base finish and have it redone...There is a company in the St Louis area that specializes in refinishing Brownings and they come out looking like new...I'm sure you can Google it and find it, if not, I may have the name and number available with a couple of phone calls, but I would have no idea as to the price...
I haven't looked, but there may be a forum for Brownings on the 'net that may be able to provide some specific information on the problem..
At one time I had a BSA spotting scope that had a similar finish on it and without applying anything, it became gummy in the plastic storage case and I wound up rubbing off the "All Weather Rubberized" coating just to get it where I could handle it...It took a lot of elbow grease and alcohol...At least now, it can be handled without applying a lot of black on your hands...