Migration is often economically motivated, and it seems that Appalachia has been a culturally distinct and historically poor region. I was curious whether the Appalachian people have been a part of any significant "waves" of migration to other parts of the United States analogous to the "Great Migration" of African Americans.
Sure, and at around the same time as the Great Migration of African Americans. When Henry Ford began offering $5 a day for working in his factories and the midwest economy began to boom,a lot of people in the Appalachians left for good jobs in Chicago, Cleveland, Detroit. One route, Highway 23, was called the Hillbilly Highway. Here's a little article on one enclave that still survives in Detroit.
It is a major theme of music and writing of the region. One of the more famous Appalachian novels, Harriet Arnow's The Dollmaker, concerns a family moving from Kentucky to Detroit, much as she did. Few people notice that the two most famous songs about West Virginia, Danoff's Take Me Home, Country Roads and Hazel Dickens' West Virginia You Are My Home are written from the standpoint of someone who doesn't actually live there anymore. Danoff was just a songwriter from MA, but Dickens actually moved from WV to Baltimore fairly early in her life.