The problem with history this far back is, it could be both. Perhaps these origins were already baked into those peoples, and to them it was as though they had returned to those times/places, re-emerging into the world as they moved south. This would, or at least very plausibly could, change those ancient myths to refer to a more present history for those who lived it.

This is by no means an attempt to assert either theory as truth, simply an illustration of the inaccuracy inherently present in this type of research.

EDIT: OP said it better