That doesn't rule out a Balkan language showing up, but it's more likely that Balkan language would have already been transplanted (along with the women) to the East by the Persians, and then again by Al Iskander.
So, a double-whammy.
Rather like Hungarian ~ which has several major roots. The Dravidian component had to have been transplanted to a number of places first because that is one big hike! Or sled ride, or maybe donkey trip, or goat wagon trek!
The idea that the Uralic or Finno-Ugric languages (Hungarian, Finnish, Estonian, and others) and the Dravidian languages have any connection seems to be a minority view. The homeland of the Uralic languages and the homeland of the Dravidian languages, even assuming the latter were more widespread before the Aryan invasion of the Indian subcontinent, are pretty far apart. The only distant language which is sometimes thought to be related to Dravidian is Elamite, and that seems to be unproven.