Cage's use of Irish music in Roaratorio is pretty incidental, and it was planned out before most of those players were even born. Basically it's a response to Joyce, i.e. the highest of high culture.

I take Jerry's original question to be about what happens when a group of people takes some kind of music from outside to its bosom and makes a tradition out of it. There is one elephant-in-the-room example of that: religious music. Whether it's Catholic chant, Protestant hymns or Islamic calls to prayer, most believers end up performing music from the tradition of the culture their religion came from. What happens next is the interesting bit. Gregory the Great, Luther and Mohammed couldn't have predicted Corsican folk polyphony, Hebridean psalms, or Sudanese Koranic recitation.