does it use much oil?
I'm guessing that the old pipes have carbon from old engine?
Humidity going through engine is cleaning old carbon out of pipes?
I don't know. How much stuff on floor are we talking?
Could be an extreme rich condition at start-up. The choke could be causing it, or the carburetor could be improperly tuned.
I've had this happen on a new rebuild myself and the culprit was poorly sealing rings on the pistons. The rebuild was no good.
If you're trying to find a culprit, I would look in the obvious areas first, such as, is it blowing smoke out the pipes (either black or blue) during these start-ups or at other times? Are the spark plugs fouling or have any sort of build up on them? Clean burning will leave a nice golden brown (not glazed-over brown) on the ground insulator like freshly baked bread.
If there aren't any other indicators of problems, its normal to experience drainage out of the exhaust in high-humidity environments and as has already been mentioned, if there's carbon build up in those old pipes (or the engine's combustion chambers, since it may not be driven enough to clear them out) that humidity will steam clean that stuff out of there.