I would like to think the smoke is being pulled down into the hotter part of the fire and combusted, making it cleaner.
The smoke is mixed with fresh air above the fire and combusted, making it cleaner. You can see it happening.
Thinking about the jacket and simplifying the model by losing the holes in the outer jacket. The air in the jacket is being warmed and will expand, becoming lighter than the air outside the jacket so will exit the jacket at the top. This leaves a low pressure area which can only be filled by air either coming back down the jacket or being drawn throught the fire. The former I suspect would pulse, the later give the draw down effect described.
Yeahbut:
The air in the pot is being warmed *more* and will expand *more* and rise, this leaves a low pressure area which can only be filled by by air coming back down the jacket. This air will contain rather a lot of combustion products, and it will either reach a smouldery equilibrium or go out (indeed, you'd have the same problem if it were being drawn down through the fire).
Of course this is buggered up by the big holes in the outer jacket which will allow air to be drawn from the outside up through both the jacket and the fire.
Which is how they seem to work in practice. The experiments to do are to try running one without the jacket in place, and to draw off a sample of the air inside the jacket and test for combustible gasses.