Improvising variations - known as 'Divisions' - goes back many centuries. Classical musicians would have been expected to improvise to a 'ground' or bass. I think it's the same principle.

In collections like the Thomas Marsden, the tunes all have written-out variations, which I gather was unusual, and the fact that the Marsden title page boasts that it contains Divisions on each tune would support that. Some of the suggested variations are good, some are interminably banal.

So I don't think Northumbrian practices of improvising variants comes from jazz. Wouldn't the timescale be wrong anyway?

Improvisation aside, I do find, however, some melodic aspects in common between some Northumbrian tunes (especially the show-off hornpipes) and early jazz. Some hornpipes sound a bit ragtime. Ones that have a lot of chromatic runs in. Things like 'Flee like a Bird'.

Although I'd say this is coincidental, it nevertheless seems like synchronicity: in the late 19th century you hear a fair bit of jazzy turns of phrase in American folk tunes and also in classical music. I think it's perfectly possible that different musicians thinking and playing different types of music can be having similar ideas at similar times, without ever hearing each other or being directly influenced by each other. Certain progressions of an art form just seem to be in the air.