1) the main pulse of Indo-Europeans, the proto-Greeks, arrived ~2300 BCE to “mainland Greece” (i.e., the north). This notwithstanding other earlier contacts noted in the text between the Pontic steppe and the Balkans2) The Minoans and other peoples of the Aegean did not have this ancestry. This is not surprising. But, this works seems to confirm a likely pulse of ancestry into the Aegean ~4000 BCE with roots in eastern Anatolia and/or the Caucasus. This is a minority component, but seems correlated with the arrival of Y chromosomal group haplogroup J2, and has been detected as far west as Sicily.3) The above component is related to the contributor to about half the ancestry among the Yamnaya samples. But, the Yamnaya samples themselves are about half “Eastern Hunter-Gatherer” (EHG), which itself can be decomposed as 25% “Western Hunter-Gatherer” (WHG) and 75% “Ancient North Eurasian” (ANE). This EHG component was lacking entirely in the Minoans of the Bronze Age and is lacking in modern Cypriots (who are mostly ethnically Greek). In contrast, the EHG component begins to increase in the Balkans during the late Neolithic.4) There seems to have been a further dilution of the steppe component among the Bronze Age Greeks as they moved from the north to the south. The largest component of Greek ancestry then, and now, remains “Early European Farmer” (EFF), related to and descended from “Anatolian Farmer” (AF).5) Modern Greek samples have more steppe than late Bronze Age samples (Mycenaeans). I am confident this is due to early medieval Slav tribes, who moved as far south as the Peloponnese in large numbers. I’ve looked at a fair number of Greek samples, and some of them have way less steppe ancestry than others, with the latter matching those labeled “northern Greek” by the Estonian Biocentre dataset. I think many of these former are likely island Greeks from the Aegean or Greeks who descend from early 20th century migrants from Anatolia.
So, basically, you have a four wave model:
1. Anatolian Neolithic migrants.
2. Copper Age/Early Bronze Age migrants from West Asian highlands.
3. Indo-Europeans from North to South and petering out as they move south.
4. Medieval Slavs (who are Indo-European) also on a North to South cline.
Davidski at Eurogenes confirms the likelihood that there was a wave of Slavic migration to Greece along these lines but is more skeptical that we have good enough data to make strong conclusions about the overall population history.
The West Asian highlands wave largely didn't make it into Europe (apart from Sicily, a few other places in Italy, Malta, European Turkey, and Greece).
The presence of this substrate in the places where the Anatolian languages emerged, and its absence elsewhere, is, in my humble opinion, what makes the Anatolian Indo-European languages (most famously Hittite) so divergent, despite the fact that it is really contemporaneous with Mycenaean Greek, Sanskrit, Avestian Persian, Tocharian, and other European Indo-European macro-language families.
This West Asian highlands wave also tends to support my hypothesis that there is a macro-language family associated with it that includes Minoan, Hattic, Hurrian, Kassite, and probably some of the Caucasian languages, in addition to providing an Anatolian Indo-European language substrate.
I'd also guess that at a time depth too great to construct using standard linguistic methods, that all or most of the Caucasian languages, Sumerian, and Elamite are also part of this macro-language family, for which ergative grammar is probably a good litmus test.
This data and Razib's analysis also suggest the possibility of using Y-DNA J2 as a possible tracer of this language family's range in cases where subsequent languages acquired by language shift can't be ruled out, and to narrow down which Caucasian languages were mostly likely associated with this wave (most likely the Northeast Caucasian languages like Chechen and Ingush, where Y-DNA J2 present in a majority of men who speak those languages).
If Y-DNA J2 is a reliable tracer of this wave, then it also suggests that Basque and the extinct Vasconic languages are probably not associated with it. See, e.g., here. This thus favors the alternative hypothesis that Basque is derived from the language of the first farmers of Iberia, probably derived from the languages of West Anatolian early Neolithic farmers, rather than languages related to, for example, Minoan, Hattic and Hurrian.
Indeed, quite likely, both the Western Anatolian language family of the Neolithic first farmers, and the West Asian highlands language of the early metal age were remote cousins within this macro-language family, with Y-DNA G (associated with the first wave Anatolian farmers), J1 and J2 tracing deep historic outlines of some of its main branches (with J1 eventually undergoing language shift or language evolution to Afro-Asiatic languages like Arabic, Coptic, Berber and the Ethiopian languages).