The Greek metempsychosis seems to have been quite similar to late Vedic and Upaniṣadic reincarnation and like the late Upaniṣadic thought promoted a liberation from that cycle. We first see this appear in the late 8th or early 7th century with the emergence of Orphism and Greek mystery cults and later see the doctrines expanded in Pythagoreanism and Platonism in the sixth and fifth centuries respectively.

Indeed much has been made of the similarities between these underground Greek traditions and the Religion of India, not just in their theories of reincarnation but also cyclical thought, ritual, monasticism, and ascetic practices. For instance, the Pythagoreans were monastic, vegetarian, white-clad, and likely celibate, much like the Jains and likely their earlier ancestors. The similarity in practice and basic doctrine (although the related cosmology is very different for all traditions in Greece and India) seem to suggest a common origin.

Some have suggested that it is the common proto-Indo-European root of their religions, but others (including myself) believe this unlikely. There are two issues with this theory: 1) there is little to no evidence of reincarnation in other derived religions, and 2) reincarnations absent or poorly developed in early Vedic texts. This suggests that this was not a feature of the early Proto-Indo-Europeans or indeed any of their succeeding tribes like the Proto-Indo-Iranians or even the early Indo-Aryans.

Several explanations have been put forward to explain the bizarre near contemporary appearance of such similar doctrines over such a large area. One is a near eastern influence from the Babylonians or Assyrians, which is unlikely due to it not being seen in any texts of their well attested religion. Another is via Egypt which is supported by Herodotus and possibly readings of various versions of the book of the dead. Yet another is diffusion from India via Persia, which was a cosmopolitan center of trade, science, and philosophy following in the mid iron age, either developed by the Indo-Aryans or adopted from the Indus Valley Civilization or other native inhabitants.

All of these have supporters and detractors, but I personally believe it to have been an Indian belief diffused through Persia (where many early Greek philosophers were said to have gone, including Pythagoras and his teacher), though Egyptian influence cannot be entirely dismissed.

I will add sources soon and if you'd like can post a detailed breakdown of beliefs and doctrines according to various traditions as well. Feel free to ask questions to clarify.

Sources

McEvilley, Thomas (2002) Shape of Ancient Thought

Banwari, Siobhan (2015) The Theory of Reincarnation and the Journey of the Soul

Long, A. (1999) The Cambridge Companion to Early Greek Philosophy

Hutchensen, D.S. (2017) Protrepticus

Khan, Charles (2001) Pythagoras and the Pythagoreans

Guthrie, William (1952) Orpheus and Greek Religion

Linforth, Ivan (1941) The Arts of Orpheus (No Relation)

Various Primary sources in translation and things that I can't think of off the top of my head where I got this info.