The "folk process" is simply a term that refers to the way that a community uses its available resources to adapt something(in our case, music, but it doesn't have to be) to its own purposes and tastes.

Broadsides, sheet music, and recordings show us how a piece starts out, but folklorists and ethnomusicologists are concerned with the way that communities change it to suit their needs.

The study of the "folk process" is very much akin to the study of language changes, and it is worth noting that the departments of linguistics and folklore are very closely associated as schools such as the University of Pennsylvania.