Took it to browning and they said it needed 300.00 worth of parts. phooey! I rifle is in perfect condition. Part of the recommendation was to refit and rebed the forearm. That makes a little more sense. What do you guys think and do do you do the job. This is an old 1969 Bar I got from a texas feller so the wood might have shrunk a little, just guessing. The first shot is at point of impact and each succeeding shot climbs up and inch or two. It's fine left and right. Never got the bbl hot shot slow on a cool day. Appreciate your imput.

I have a fairly new BAR that shoots the first shot to point of aim, the second very close by (a little higher, though) and the third one still higher. If I continue, it continues to climb, but I usually don't because the barrel is too hot to hold by then. I think that it is the nature of the beast. The hot gas is tapped off about halfway down the barrel and piped back to operate the action. Differential expansion from the connected parts could easily shift the point of aim. If this were a varmint rifle, I would be very unhappy, but I can live with that in a deer rifle. If you find a solution, let me know.

My guess is the same as yours: Differential forearm pressure on the barrel as it heats up with successive shots. Makes sense and I've seen a number of lever guns do this, although I've never had any bedded to correct it so I'm just guessing. For a deer gun it seemed more important that the first shot go where you aim and if the second is an inch or two higher than that it's not the end of the world and you'll probably still hit the deer.
-WSJ

You've already got the problem figured out. Fix the forend. It's touching somewhere and getting worse as the barrel heats up.