Has anyone considered building a rocket mass heater where the thermal mass is something with air space (like brick rubble) in an insulated box? Then you could hook a thermostat up to control a fan that draws air through the thermal mass and into the room (or even another room via duct). It seems to me that this could "stretch" the heat out further while also keeping the room where the RMH is from getting too hot while its burning.
Sounds a lot like Paul's "pebble style" mass heaters.
Bit of a conflict between thermal mass heat storage, and air gaps that rob the heater both of conductivity and of stored heat.
Converting overnight warmth into more immediate air warmth seems like a waste to me - you are moving back towards the problem of having to run the fire all night to stay warm.
Air does not store heat well. Also, buildings with live creatures in them (such as people) need to cycle a lot of air to stay livable, and that means air-heat is soon lost heat (about 1/3 of the building's volume per hour, minimum).
For spaces that are not occupied all hours, where heat is needed intermittently (like a big church hall or something), I could see that direction being worth pursuing.