We tried a few methods and eventually went to pair programming. But never tried the ping-pong pull requests. It sounds interesting. What gets me off for pair programming is that it's not always feasible. If the other person who could work with you is on another task you spend a lot of time try to play catch up. How well did the ping pong work?
It works well. I think with the original pull-request process we basically had a mental block that only one person is allowed to write the code, and the reviewer can only comment. This can be frustrating especially if you know exactly what needs to be fixed. If you just go ahead and make the change, you save both people's time.
I guess one place where it fails is that it's not very educational. If you're tutoring a less experienced programmer, you don't want to fix the mess after them, you want to teach them how to doing without you the next time around. In that case, I either leave the fix to the author, or fix the issue as they watch.