I have used water down the bore since 1988, cools the barrel in less than 30 seconds with 6 oz of water. Follow by 4 tight patches on a pointed jag, dry chamber, back to shooting in minutes.
All the guys I took p. dog hunting used the same method, on the very finest custom Varmint and benchrest rifles made.
Another method that will work, but not anywhere near as fast is to drench a wash cloth with a mixture of 50/50 water and rubbing alcohol. Rub the barrel with the wash cloth till the solution is dripping off the bottom of the barrel. The solution will rust a chrome moly barrel if you don't oil it when you get home.
Don't be scared of putting water down the barrel, we have done it for more than 100,000 rounds fired.
The whole idea of compressed air is a bunch of BS once you have used the water down the barrel. Also, when you patch out, you remove more carbon that you would imagine and extend your shooting string, also.
Before we got into cooling our barrels, we would carry a minimum of 6 rifles each and fire 50 rounds per gun, the barrels never really cooled off....no surprise in 90* heat.
For developing loads at the range, the 50/50 rubbing alcohol/water works great, cheap and easy. Most factory barrels have their groups open up once they really get hot.