Keep in mind, I'm grossly oversimplifying it. The linguistics bit is pretty solid, based on common roots and such. And biologically, the migration stuff is really solid. Recent research suggests that pre-modern humans were more mobile than we might suspect, but the 'story movement' idea also has some far fetched necessities. Someone would have to travel really far, hear a story that somehow applies to their life, remember it, travel back, find the story interesting enough to repeat, and then tell the story to enough other people with similar circumstances that the story travels not just in a small region but across an entire continent. The claim is more in regards to the basic structure of the story, not exact details. Honestly, common ancestry sounds better to me. Human oral history is a pretty amazing thing. Homer's tales were oral stories for at least hundreds of years before he wrote them down. Bulgarian storytellers can reach back a similar distance for some stories. Lasting a couple thousand through a much larger population doesn't seem that far fetched to me.So that's total bullshit.
I might be biased because I recently learned about the "paying with sound of money" story (If you read french you can read about it in one of my blog) Its roots are unsure, but around the XVI century we had the same story in Arab, in German and in French: The hero (Nassreddine in Arabs, Till Eulespiegel in German, Joan in French) is hungry. On the market he sit next to someone selling pie (or chicken) and spend the day enjoying the smell. The Merchant ask to be paid, and the hero pay him by making noise with money in his purse. It make more sense to assume the story traveled for one to the other place than to guess it was already known when German/french/Arab was one tongue. My bias is that story travel fast when they are good. My Occam Lightsaber guess a story would have been written by someone, somewhere, before the Grimm Brothers in the XVe, if it was out for 4000 years. But may be the science in the common ancestry theory is sound. I dunno, Linguistic is far out of my reach.