Engine cutting out

No spark/replaced coil.  Are you running conventional points and condenser or Electronic Pertronix Ignition?  Pertronix have been know to fail under heat condition.

https://itstillruns.com/do-tes...gnition-7993715.html

It's got to be as hot in TX as NW FL last couple weeks!  There was a thread here to build "a prop" that keeps rear lid open a few inches for extra air. 

Is the engine properly grounded? Sounds like something is loose.

Chuck that Pertronix as far as you can throw it. 

My take as well. Bad ground or connection or faulty Pertronix.

x3. Sometimes a new coil is bad too, but that's a rarity.

Ground, Pertronix, coil. ... BTW In the last few builds I did I ran into a number of bad electronic ignitions and went back to good Bosch points .

@Chas

I had the same problem a while back Chas. I replaced my Bosch 009 with a Maganspark electronic distributer and the daypack coil. I could drive the drive and it ran great but after a little bit of driving the car would sputter, spit, and stall. It wouldn't start right away but if I waited a little bit I could start it, drive a little, and repeat the process.  

Turned out I got a bum coil. As it heated up the spark that travels through it would get interrupted (possibly a broken wire inside) and I'd have troubles. I replaced the daypack coil with a new dry pack coil and it's been running great ever since.

Ignition issues are across the the board as for replacement points, condensers, cap and rotors are not the quality they once were. I had points close on a short test drive after installing them. The friction material on the points is not of the same as oem parts were years back.  I also had issues with just about every electronic unit too. Recently had "no spark issue" and the red wire in the electronic module was loose and fell out. As for the coil failing I see more of that too. They get so hot too touch is a sure sign.

i am still partial to a set of points.

When I was doing my build, I remembered my old Mustang, and on that the coil was mounted on a inner wheel well, away from the heat of the engine, so that's what I did on my Speedster.  The Magna-spark coil, a big brick of a thing, is mounted on the driver's rear inner wheel well partition where it is a lot cooler than if mounted on the fan shroud.  Sure, I needed a longer coil to distributor cap wire to reach, but that was no biggie and everything has been working now for almost 20 years.

I with the other guys, though......Chuck that Pertronix and either try another one, or a different maker or go back to points.

Those symptoms remind me of my 69 VW bug... culprit ... the coil  

Gentleman, thank you for all the feedback.  I will return and report what I find once I dig deeper. Initial attempt was to replace the coil as instructed here by many.  In doing so, the same behavior occurred with the 2 additional coils I installed.  So, I'm turning my attention to the distributor as well.   More to follow...

Tell us what's in there for "points" now and what brand coil you are using.  

From Rob and Dave's site-

If you are running points, or a points replacement device (Compufire or Pertronix), you MUST make sure your ignition coil has the ballast resistor. Stock Bosch blue coils have the 3 ohm ballast resistor INSIDE the coil. If your coil or primary ignition winding doesn't have this ballast, the distributor will burn up and leave you walking. There are five different "Bosch Blue Coils", and only one of them has the ballast in it.

The high-performance electronic coil is designed specifically for use with electronic ignition (e.g., Compufire, Pertronix) which deliver brief high current pulses. If you are running standard points ignition, stick with the OEM coil.

My magna spark 2 with the dry coil had issues as well.  Car did the samething you described.  Replaced coil and solved problem.