I've been chasing this very same problem now for a while, thought I had it solved. The symptoms are the same, and as I understand there could be multiple causes. I replaced the fuel pump and fuel pump push rod, thought it was solved, and it was for a while. Then I replaced the electronic ignition box and ballast resistor, thought I had it solved, and it was for a while. Then this weekend on Saturday night after it cooled down outside a little bit (It was 90+ degrees here this weekend) I decided to take the Road Runner out for a drive and fill up the tank for the trip over to the show on Sunday. Of course I took the long way to the gas station and the extra long way home. And to be honest I was pushing it on occasion, what fun is it if you don't, right? I don't have a temperature gauge, just the one with the needle on the dash. While sitting at a red light about 1/2 mile from my house and the gauge creeps up while sitting there to about half way. So it was a little warmer than where it is when running, which is usually around the 1/4 mark or slightly higher.
The car starts to choke a little bit, so I put it in neutral and the idle goes back to normal. Then about 10-seconds later it shuts off and won't start. F#&%!
I get on the phone to call for help and the Cops come. They were nice enough, sympathetic, and helped push it off the road so traffic could get back to normal. As I'm waiting for the tow truck to arrive, I get in the car and try to start it, it starts right back up! I drive it home and put it back in the garage like nothing happened. Called and canceled AAA.
That Sunday is our Mopar clubs car show and I say screw it, I'm going, I don't care what happened yesterday. It's about a hour drive and I'm supposed to meet a new guy with his first Mopar, a 1970 Plum Crazy Purple 383 Challenger. Nice car, but its a project, but a solid bones running car that could be left as is, or restored. Anyway, I make it to the show, another hot day, lots of red lights, but no problems for either of us. While there I'm talking to a fellow bird owner about what happened the night before and he says "The same thing was happening to me, follow me" and he takes me over to meet this other guy, who is a mechanic and corrected the problem on his car. He proceeds to tell me it sounds to him its what he called the "Distributor clip". I stare blankly off into space as I do most times when things mechanical related to engines are being discussed. But he said you take off the D-Cap, and it is under the Rotor. If it goes bad, and apparently they do, if the car gets a little hot it will cease to work and the car will shut down until cool.
I made it back to the garage after the show that afternoon, no problem, really hot again over 90 degrees outside on the ride home. My temp gauge is holding at 1/4 to 1/2 way over depending if I'm moving or not. We're about a 1/2 mile from the garage and glance in my rear view and the Challenger is no longer on my tail. I call Ruben, and he tells me his car just shut off at the bottom of the hill, so I get in the Jeep and head down to at least be moral support and discuss what to do. His car has an after market temp gauge on it and it was over 200, but its not spitting water or anything. He is pulled over in the shade, I tell this problem sounds familiar and we decide to lift the hood and wait for about 30-minutes and let it cool down a bit, and the car starts right back up like nothing was wrong.
I'm going to do some research on this distributor clip thing and make a few calls. Any input here would be appreciated as usual.