Jonty,
I have an old Beretta ASE manufactured in the late 1950s in 98% original condition that is doing the same things as your Browning. As with your Browning, the barrels and ribs on this ASE are solid and ring nicely, but when I get the barrels really hot I have a spot that weeps a barely noticeable bit of moisture.
I do not have a warranty... so I guess I just have to bite the bullet and find someone to do a repair.
NCTarheel