When we had our 22' fifth wheel, the passenger front tire would always lock up but the others kept rolling... the problem wasn't necessarily the wire size, but how it was wired. The factory routed the wires along the right side of the trailer, then tapped off and ran across to the left side. The right front tire was first in the circuit.
I rebuilt the wiring using a variety of sizes. As Larry Cad stated, each brake pulls around 3 amps at full braking. 3 amps is easily carried by 16 gauge, which is the same size as provided on the brake puck. I wired each of the 4 brakes to an equal length 16 gauge pigtail. At the center of each axle, I joined the pigtails to equal lengths of 14 gauge, then ran those back to the right frame rail. Those, then were joined to a 12 gauge lead that ran to the 7 pin plug.
All connections were soldered, sealed with liquid tape, then protected with heat shrink.
The point of this exercise was to provide equal current and voltage to each brake. From that point forward, I never slid one wheel, but could lock all four up if desired. It was a vastly different towing experience.
All that said, the point isn't necessarily just the wire size, but how its wired. You don't need 10 or 8 gauge wire to each brake, that's overkill. But you do need to consider both the total current on each lead and how it's balanced.
On the tow vehicle, you'll need an adequate size wire for the job. At 12-14 amps maximum for 4 brake trailers, 12 gauge (20 amp max) is all that is needed. In practice, you'll never pull 12-14 amps on a constant basis. I'd venture a guess that current during usual braking is commonly between 1 and, say 6 amps. Those who have current readouts on their brake controllers know the story.