The information I linked to is the best you're likely to get just now, though Walrus adds some useful additional stuff.  The tune is, as I said, Ça Ira, renamed in England The Downfall of Paris and subsequently taken up in Ireland -with a hornpipe rhythm- as a setdance tune, and in America as a fife piece; a march again.  Mississippi Sawyer clearly derives from it.  As per my link, the Paris bit comes from Shield's opera of 1790, and refers to the Trojan prince, not to the city.

The Internationale is completely irrelevant to this discussion.