Happend to me 4 times with the DJI RC PRO Connected to the DJI Mavic 3 Pro.During midflight, the RcPro initialized a system shutdown. Blinking Red light indicating a possible overheating issues.
The RC is in front of a portable electric fan and I also tried flying inside the car for air conditioning purposes but it still initiated system shutdown.
I tried to use it with the DJI Mavic Mini 3. I got no overheating issues with it. I hope this can be fixed with a hardware firmware update.   

All firmwares are updated to the very latest version.

Inputs are appreciated and I hope DJI can help me solve this overheating problem.

I use the DJI RC Pro with both my Mini 3 Pro and Air 2S in the hot Florida weather and have never had it to overheat yet.  Must be something to do when in use only with Mavic 3.  However, as you said it has happened 4 times it might be a defective controller.  

Do you hear the cooling fans coming on?  Maybe they are faulty?

Hi, Aj. Thank you for reaching out. Please refresh the firmware of the RC Pro using the DJI Assistant 2 (Consumer Drones Series) which you can download for free from our website (https://www.dji.com/downloads/products/mavic-3). If the issue persists after refreshing the firmware, please provide us a screenshot of the error message (which shows initiated system shutdown or overheating prompt) for further checking.

Hello, there, Sorry for the inconvenience. May we know which aircraft you connected to has the heating issue? DJI Mini 3 Pro or DJI Mavic 3?

We are sorry that if it is convenient to you to export the LOGs in DJI RC Pro for us?
Our engineers would like to further analyze the issue.