I played around with my two 600EX-RT and one 430EXIII-RT during Christmas, had one 600EX-RT on the camera and the two other flashes linked and shooting with 100%. I haven't counted the shots (likely more than the maximum of 20 Canon recommends to do in 10 Minutes), but at some point everything slowed down because the 430 went into overheat mode. The 600EX-RTs were doing fine when I removed the 430 from the setup.
The 430EXIII-RT definitely overheats faster than the 600EX (with the 600 having about twice the power).
I originally intended to perhaps get more 430EXIII-RTs (two of them cost about as much and have roughly the power of one 600EX-RT, but you can spread out the flashes if you want to), but with the overheating I think if I need more power I will go for more 600s.
I'm not completely sure if the 430EX II ever overheated for me. Once it was slow to recharge which might have been the overheat mode, but I assumed it was the batteries and I just replaced them and it was functioning normal afterwards. But I haven't shot much with that flash, because I quickly replaced it by a 600EX-RT because a flash is never powerful enough.
In the manual Canon gives an estimate of 32 shots at full power until the 430EX III goes into overheat mode. Seems to be roughly accurate for my observation. 8 Flashes is really low. Did you fire the flash before the 8 shots sequence? Was the ambient temperature higher than normal?