"So, once it begins it just keeps going even after the intrusion of water has been eliminated? What feeds or sustains the process thereafter?"
Moisture , wood rots at only specific moisture levels.
The disaster of a leak is that the wood will pass thru the zones getting wet then drying out.
Seal the leak , and the rot is still there and will continue to grow as the the moisture changes.
Cure , chop out ALL the rot to regain some structural integrity , and use Cuprinol or similar to slow the next rot infestation.