The easiest way to ifx that barrel walking on a BLR is to glass the forened to the barrel so it is making complete and even contact all the way .
Some guys have gone as far as actually permanently glueing the forearm to the barrel using Devcon or other bedding epoxy but I have not found that necessary on the ones I have done. Not all models of the BLR have a barrel band but if yours does I would ditch the barrel band. When you are applying glass make sure you glass the back of the forearm where the forearm butts up against the action.
It is very easy to do, and instructions on how to glass bed a rifle are plentiful if you google it. Brownell's site has a very good starting explanation.
You don't say whether your BLR is a takedown model or not. If it is a takedown and the glassing of the forearm doesn't cure your problem it is also possible you don't have a good tight fit of barrel to receiver and the heat is moving the alignment. This can only be fixed by Browning or a VERY good gunsmith.
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