According to the Biblical record, the Elamites appeared to have started in Mesopotamia (early record shows that they attacked the city of Ur just after Abraham left;, and then moved into Persia / Iran.
However Jeremiah the prophet (~600BC), then prophesied that they would be scattered in 4 directions:
(Jeremiah 49:36 And upon Elam will I bring the four winds from the four quarters of heaven, and will scatter them toward all those winds; and there shall be no nation whither the outcasts of Elam shall not come.) The Elamites were later overthrown by the Persians.
Yet, somehow "restored" in the "latter days":
Jeremiah 49:39 But it shall come to pass in the latter days, that I will bring again the captivity of Elam, saith the LORD.
Many scholars assume that prophetic Elam, still refers to Persia / Iran, but given that they were scattered and later to be restored, unless there is a relocation of Elamite descendents to Iran, it is possible that this restoration is elsewhere.
The one group today that has a number of apparent links to the ancient Elamites, are the South Indian Dravidians. If so, it
represents an Elamite group that moved to the Indus river (modern day Pakistan) region, and then further south to their current location in South India following the Aryan invasions (as recorded in the Vedas, and matching the secular history viewpoint).
If the Dravidians are descendents of the Elamites, then they are also descendents of Shem and would share certain Semitic histories and culture. In particular, the main Dravidian deity is the moon god, Shiva. Across the Middle-East, the ancient moon-god is considered to be the deification of Shem (and the fish-god being the deification of Noah). Furthermore, the prophecy of Jeremiah 49:39 could then apply to them.
There's some interesting information suggesting semitic (i.e. Israelite, Assyrian, Elamite etc) cultural links with the Dravidians at desivoice.yuku.com/topic/2645