The first starts in 6850 BC with the Seklo culture in Greece and lead to the foundation of Starcevo (former yougoslavia) and Körös culture (Hungary) . Those very first farmers may have carry the Mediterranean/ Southern admixture as well as haplogroups I2a and G2a.Right now in another thread a new paper about iranian Y-lineages is being discussed, where Y-IJ has been found. It seems to support my theory that near-eastern immigrants were not only neolithic farmers, but some were still hunter-gatherers.That second wave of Neolithic didn't reach Western, Northern and Southwestern Europe (except Sicily and southern Italy which are high in J2) that is why western and northern european have more Mediterranean than west Aian admixture.
"F-M89* and IJ-M429* were observed in the Iranian plateau: the first represents the ancestral state of the main Euro-Asiatic haplogroups  while the second probably moved toward southeast Europe sometime before the Last Glacial Maximum where it differentiated into the “western Eurasian” haplogroup I"
Further, J2a-M92 variance distribution seems surprisingly stronger in the Balkans and Italy than the near-east, according to the maps. So maybe even J2 could have been already in Europe during the palaeolithic?
To me it still looks more like the bulk of Atlantic/Mediterranean admixture comes from near-eastern hunter-gatherers, subsequently enriched by neolithic farmers.