I've tested more than one of the gadgets that push air or C02 through the bore, but they don't work any better (and usually not as quickly) as several simpler solutions.
If the ambient temperature isn't too warm, I set hot-barreled muzzle-up in the shade, with the action open. The convection of the hot air inside the bore sucks cooler air from the breech up through the barrel, without having to use any gadget. When barrels get really hot, pouring cold water on the outside of the barrel works as well as anything I've tried. Tilting the barrel downward water to trickle along its length. The only trick here is to bring plenty of cold water.
The other solution that allows shooting longer before the barrel heats up is to use smaller cartridges, since the majority of barrel heat is the result of burning powder. Probably 90% of my prairie dog shooting is done anymore with the .17 HMR, .17 Hornet and .22 Hornet, either standard or K. Unless the day's over 80 degrees their barrels don't heat up very fast, and the .17 Hornet is good to 300+ yards in normal wind conditions.
