I was about to post the same suggestion. I've build a RMH before the winter. Works great, but after a couple deconstruction, I can see there is soot building up on the exhaust pipe, and on the barrel top & inside. I'm burning pine mostly, so I guess this is created at the beginning and end of each burn.
Anyway, I though about the "exhaust back into the intake" in a way turbo chargers work on vehicles. Apart from the concept, it is probably very different, as turbo are also adding pressure to the intake, and it is a 4 cycle system, as opposed to the RMH which is kind of a 1 cycle ; continuous system.
But I still imagine 2 ways the exhaust gaz has to increase efficiency :
1) The residual burn gazes that still have a "energetic potential" (only in the beginning, when RMH is not very efficient)
2) The gazes are relatively warm, at least 30 Celcuis so it would not cool the burn as much as it does when the air is taken from outside. And from what RMH is about, the warmer the burn, the more efficient it is...
Even if a properly built RMH, once heated up, is supposed to be very efficient, saving some emissions in the first 20 minutes is something.
But like some said, maybe replacing some oxygen by exhaust gazes would make the thing worst...
If anybody tried, please feedback.