I have to tell you something, in my humble opinion you need to LIMIT the heat instead of welding like it don't matter and putting a wet rag or putty or what ever to keep the heat in one area. LIMIT the heat AND manage the heat by welding shorter welds and letting it cool COMPLETELY before welding more is my method.
Again, this is my opinion and how I manage my heat when welding where there is a concern. One mistake I feel that is often made is the guy will turn DOWN the heat thinking he is keeping it cooler. When in fact, if you turn down your heat (wire speed or voltage) you need to weld slower to make it penetrate into the parts you are welding together, THUS, you are heating it even more! NOTHING will heat up metal in the surrounding area more than a slow......trying to keep a puddle weld.
On the other hand, you turn the heat and wire speed UP and the second the wire hits it's penetrating and you are welding quick, you are going to have less heat spreading out. I may be wrong in a literal sense and if I did a test I may find the same heat results, but I have worked like this for years and it sure seems to keep the heat down in the surrounding areas. If we are talking sheet metal, you can weld these quick "Burst" welds letting it cool between welds and you can weld all day long on a big old piece of flat sheet metal and not warp it.
A quick weld then letting it coooooool long naturally is the best way I have found to weld with a MIG.
One of the biggest mistakes a guy can make is weld up a long seam and not have anything to do while welds are cooling. For goodness sakes don't sit there and stare at the welds while they cool, you will ALWAYS push it welding too soon. Find something else to do, God knows you have 54872 hours work on your car ahead do something ELSE while your welds are cooling. Say you are welding in a patch panel on the bottom of your drivers door. Weld here and there on that long weld welding as far apart as you can so the heat from one doesn't add to the heat from the other. Weld a foot or more apart at the closest. Then go and sand on the front fender or shape some metal for your firewall, then after you have forgotten you are even welding something go back and check to be sure it is COMPLETELY cooled, weld a few more short, fast, hot tacks and leave her to cool while you go back to the firewall or making the seat bracket or other project, then come back and feel your patch panel to be sure it is COMPLETELY cool and do a couple more of those short, fast, hot tacks and walk away going back to the fire wall....... I have literally taken hours upon hours welding something like that.
That is the way I have personally done it for years with good results at least.
If you are gas welding that is a whole different story as you need to weld then hammer and dolly the weld and you are managing the metal as you go, MIG welding is very different.
ANYTHING that is force cooling the welded area like a wet rag is likely going to cause distortion being you are SHRINKING the metal when you force cool it.
After all, THAT is how you "shrink" metal. You heat it up until the molecules are "dancing" around in it, they are spread apart from one an other "dancing". Heat it a little more and they spread apart so much the metal melts. But before the metal melts and the molecules are "dancing" around you cool it with a wet rag or air blowing on it and the molecules quickly gather close again, only when you do this they get even closer together.....shrinking the metal.
So if THAT is how you shrink metal, welding and then cooling too fast is doing JUST that. And when you are welding two pieces of sheet metal that are the shape you want them, you DON'T want to shrink it. So cooling too fast or forcibly you ARE going to shrink it.
DMB, I know you are experienced and don't FORCE cool it but to someone who doesn't know that is exactly what he is likely to do. You probably are welding similar to what I have described and your wet rags are more of a precaution or a little help for the natural cooling.