There was an excellent programme on Mummering on CBC Radio a while back, it is available from it's originator Chris Brookes here.
There is an interview with Chris here.
From my memory of the programme there was a reference to an "Obby Oss" being brought over on a very early ship, this was in their opinion for mummering, although Obby Oss type customs are often associated with Morris Dancing too. A bit off the main topic but could yield fruit if you follow the links.
I don't personally think that Morris had much purity of tune origin and used many popular tunes of the time (whatever time that was)
JohnB