The recent paper "Ancient DNA from European Early Neolithic Farmers Reveals Their Near Eastern Affinities", Haak et al, suggests that early LBK farmers are most like populations of the Northern Fertile Crescent. From their paper, Figures 3A and 3B present maps showing matrilineal distances for two LBK populations compared modern populations:
"Mapped genetic distances are illustrated between 55 modern Western Eurasian populations and the total of 42 Neolithic LBK samples (A) or the single graveyard of Derenburg (B). Black dots denote the location of modern-day populations used in the analysis. The coloring indicates the degree of similarity of the modern local population(s) with the Neolithic sample set: short distances (greatest similarity) are marked by dark green and long distances (greatest dissimilarity) by orange, with fainter colors in between the extremes. Note that green intervals are scaled by genetic distance values of 0.02, with increasingly larger intervals towards the “orange” end of the scale."
What is evident is that Figure 3B represents an earlier, more genetically distant Fertile Crescent population than Figure 3A.
If you look carefully at the Figures, you will note that Cyprus is at a genetic distance of 0.4 from the "green island" of Figure B. By contrast, Cyprus is genetically near the general LBK "green island" of Figure A(genetic distance 0.24).
Moreover, Sardinia is at a genetic distance of 0.4 from the population at the Derenburg site(Figure B), but only 0.32 from the broader LBK population (Figure A).
The diffusion graphs of the autosomal South European Component indicate a coalescence point in Anatolia and Cyprus and a radial diffusion from that point into the Fertile Crescent. The West Asian Component peaks in the Caucasus and gradually diffuses southwestward. The variance of the distributions indicate that the West Asian component is older and more diffuse than the South European Component.
Based on the diffusion results for the K10 Dodecad populations, in combination with the maps generated by Haak et al, I'm going to assert that Early Neolithic Migrators (Haak, Figure3B) from the Fertile Crescent were a people with a primarily West Asian signature. Seven or eight thousand years ago, the first wave of expansion of Southwest Asian peoples would not have been felt in the Fertile Crescent. The expansion of South Europeans would also not have been significant. The Derenburg site likely represents these early migrators. That is the population mapped in Figure 3B.
In the ensuing millenia, a South European people expanded from a coalescent point in Northeast Anatolia and Cyprus. We can tell this by the fact that the South European component continues to diffuse from this point today. Together with the existing West Asian people, these people formed the second wave of Mideast Farmers to migrate northwestward across the Balkans and into the basins of the Danube and Dniester Rivers. Their path is hinted at in Figure 3A.
This scenario would account for the difference seen in the larger LBK population with that of the Derenburg site.
This represents a refinement to the Demic Diffusion model. It supports that idea that Near East Neolithic Farmers made a significant contribution to the genepool of Modern Europe. However, it also suggests that the movement of peoples during the Neolithic was complex and was affected by changes that continued to unfold in the Fertile Crescent.