So if you had multiple batteries...is there still a need to let the PHV3 'rest' or cool off  if needing to do some extended flying and filming?  I know on my PHV2 there have been times when the rotors where blades are attached can get warm.

Any ideal limits on how long you should be flying these?

Don't think there is a published recommendation. Would vary greatly by location. I have flown 3 batteries in a row and the motors were barely warm. I do live in the UK though.. best do the touch test and if you think its warm, give it a rest.

Same as above, I live in Cornwall which is moderately warm than wales, but no probs with 3 batteries on the trot !

I can't speak for what is right or wrong, but in my case I've flown two batteries back to back with no issues.   And I'm not just floating in one spot, I'm working the craft aggressively flying around a canyon (in 85 degree weather).  Also I'm at or above 7,000ft starting elevation so I have thinner air that makes the craft engines work harder to maintain alt.
Again, this might not be officially recommended but I've had no issues.  Unless I hear otherwise I'll probably fly 3 batteries in a row once my third arrives.  P3A.

Yes it is a very good practice to let the Motors ESCs and Main Board cool down a little bit . Around 5 minutes if you can stand to wait that long . Not cooling down and keeping the insides hot should not kill the bird , just lessens the life of it meaning it could die in midair.

I would recommend a cool down period of some type . A fan or cool dry breeze of some sort or just wait a click or two. There is a lot going on inside of that bird.

There are 4 fans blowing down on the bird to keep it cool when flying including one in the gimbal. Like others have said, do the touch test on the motors and gimbal, if its to warm to the touch let it cool. Haven't seen any issues with mine yet, 3 batteries back to back and all was just luke warm. The batteries on the other hand get quite hot and they need to cool before charging. The charger won't even start to charge until they cool down, good feature. Hot 90+ days might need some cooling rest for both bird and pilot

It's not the motors that overheat.  It's all the grossly underpowered Android Tablets that seem to be overheating & failing after a few minutes.  I have had ZERO issues with my Phantom 3 Pro using an iOS device.

Based in Singapore where the ambient is normally 35°C and it gets extremely hot whilst flying. I leave it for at least 30 minutes between batteries and even then it is still warm.

Odd, not sure what happened there.

anyway, we in Singapore where ambient temperature is 35 celsius the phantom battery and motor and whole frame get hot, I am leaving at least thirty mins between flight and the p3 is still warm to the touch. No issues yet though.