It could be different things. Fire going out, then having the fan come on, til etc., etc., could be the wall thermostat opening premature, due to way low antipator(in a mechanical stat) or too often heating cycle time setting, or temp swing controls in the stat set to low like at 1 degree.
The reason the blower can come on after the fire stops is that the heat keeps building through the walls of the exchanger, then finally reach the fan limit temp sensor and kick the fan on.
There is more it could be. But I would start here, to see if the stat is short cycling.
Marbobj,
I am only a handyman, but now have built up some years tinkering with HVAC (starting 24 years ago with various mobile home furnaces), where more years back I was more into the building aspects. I enjoy the challenge of diagnosing anything. There are more actual even licensed experts here, including mods.