Just my 2 cents as a manufacturing engineer. My information is that almost all the guns of the Parker, Ithaca, LC Smith, Fox, etc., era are assembled with essentially eutectic (ratio giving lowest melting point of the constituents) lead (Pb) and tin (Sn) (63% tin/37% lead) or very close to that alloy. This type of solder melts around 360-380F. The breach of the barrels are brazed together on a Parker using a much higher melting temperature alloy. This is often an alloy that has silver added for strength. This temperature often is in the 1000-1200F range.
Steels will begin grain growth at elevated temperature for extended periods. The higher the temps, the more rapidly the growth of the grain. Or longer times at lower elevated temps will cause this too. With enough time and temp (way beyond the temps and times needed to solder) the steel will grow (not shrink). (Shrinkage in welding is a completely different phenomenae).
I have re-layed ribs on a Parker. Mine was lead and tin soldered originally and I used this for the re-lay.
It's almost universally accepted that 60/40 tin/lead is the original and repair alloy. Tix is an indium based alloy that melts around 150F. I don't know about your use, but my barrels can approach this temperature and undoubtedly exceed it, just setting in the direct sunlight on a 100F day in California. Certainly, the barrels will exceed 150F if fired more than a few times in succession on a hot day. We've all had blisters from touching a hot barrel after some repeated shooting.
go to the link below and there's an article...sort of a "how-to".
http://www.picturetrail.com/sfx/album/view/18691676