You might have been thinking, isn't the question of where these people came from one that we can now answer with genetic testing? And, yes, the same thing has occurred to historians interested in these groups, and to genealogists from within them. But the genetic work done before now has been of dubious quality. Now, though, I can point you to a study of Melungeon genetics that seems to meet the best scientific standards. Which does not mean, mind you, that it is "true," only that it conforms to the current state of the art for the genetic study of ethnicity.
The study is by geneticist Roberta Estes and three Melungeon historians. The basic finding of their genetic studies is summed up in their work on Y chromosome DNA, which is inherited through the male line. Of the 22 "haplogroups" (lineages) they identified, 12 are European, 8 are African, and 1 is Native American. They had less success with tracing mitochondrial DNA, inherited through the maternal line, but all of what they identified appears to be European. Their tests on the rest of the DNA from their subjects, which is a vastly more complicated problem -- if you have been taken in by one of those "you are 20% African 80% European" scams, take a look at this report and see how complicated the problem really is -- seem to line up pretty well with the Y chromosome data: majority European, large minority African, perhaps a tiny minority of Native American. They found no evidence of Mediterranean or Turkish ancestry.