What is a common barrel swabbing solution using while shooting on the shooting range. I clean well at home using Dawn & hot water

There is probably as many cleaning solutions as there are shooters when it comes to swabbing between shots. With real black powder I sometimes use only a spit patch to remove the majority of the fouling. Some other solutions I've used and/or heard of is a 50/50 mix of alcohol and blue windex, alcohol and Hoppe's #9, 20:1 mix water and Ballistol and then the commercial cleaners like T/C's T-17, Slip 2000 etc. And then there are the Blackhorn 209 shooters that use a regular nitro cellulose solvent to swab between shots.
As stated above I normally use either a spit patch or an alcohol/blue Windex solution. But lately I've been trying alcohol and Hoppe's. Always one wet followed by one dry - both sides.

Are you talking about swabbing between shots or swabbing after you're done shooting, while at the range? If it is the former, my response would be the same spit patch advice as Bronko. If it is the latter, it would be to clean well, as you say you do, when you get home.

If you're talking about swabbing between shots, I use 91% Isopropyl alcohol in one of those 3oz. flip top bottles they sell for airline travel.

91% Isopropyl alcohol works great for me as well.

I keep a bottle of plain water wut a little Dawn mixed in on the bench. INE swab with tat followed by two dry patches. Same stuff at home for entire clean up. Follow up wit alcohol to remove all moisture.

(you haven't mentioned what powder you choose to use)

I use an emptied spray bottle that once housed eyeglass cleaner. In it I put
2oz - 71% isoprophyl alcohol
1oz - Ballistol.... (any water soluble gun oil can be used)

I am assuming your powder-of-choice is not Blackhorn209). Oddly enough, I have met a few different Blackhorn 209 users with traditional smokepoles over the years at nearby gun ranges.
The bore-swabbing recommendations for that powder is different.

Blackhorn 209 powder is not a good fit for sidelocks, Even the hottest ignition caps (along with nipple priming) cannot be trusted.

My muzzleloaders use Black MZ and Shooters World black powder substitutes. i don't swab between shots.

Prior to leaving the range i swab he bore with a patch wet with Windex multi-purpose cleaner.(Windex with vinegar).

At home thebore is cleaned using tap water, followed with dry patches. The bore is then swabbed with a patch moist with WD40 followed by dry patches. Lastly, the bore is wabbed with patch moist with Militec 1 or synthetic motor oil.

For my Traditions Firestorm rifle I use Traditions premoistened patches between shots, they come in a jar.

You mean a Thompson Center Firestorm.