If the earliest known performers in each city were doing the same thing at around the same time then that wouldn't support a view that it spread from one source.
If the earliest known performers were all based in new orleans and jazz in other cities only appeared some time afterwards then it is possible that it came from there and spread - but then you also need to ask who brought it ...
Lets face it - in all probability it will be much more complex than that.
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sj