I get asked about this all the time, and promised I’d post something first I heard anything, so here is a foretaste, Western Europe during the third millennium BCE: A genetic characterization of the Bell Beaker
Complex:

The Bell Beaker Complex (BBC) was the first widely distributed archaeological phenomenon of western Europe, arising after 2800 BCE probably in Iberia and spreading to the north and east before disappearing at the latest by 1800 BCE. An open question is the extent to which the cultural elements associated with the BBC spread through movement of ideas or people. We present new genome-wide DNA data from 196 Neolithic and Bronze Age Europeans – the largest report of genome-wide data in a single study to date – and merge it with published data to form a dataset with 109 BBC individuals that provides a genomic characterization of the BBC across its geographic and temporal range. In contrast to people of the Corded Ware Complex who were partly contemporaries of the BBC in central and eastern Europe and who brought steppe ancestry into central Europe through mass migration and replacement of local populations, we show that the initial spread of the BBC into central Europe from the Iberian Peninsula was not mediated by a large-scale migration but rather through communication of ideas. However, the further spread of the BBC beyond central Europe did involve mass movement of people. Focusing on Britain, which includes 81 of our new samples in a time transect from 3900-1300 BCE, we show that the arrival of the BBC around 2400 BCE was mediated by migration from the continent: British individuals associated with Beakers are genetically indistinguishable from continental individuals associated with the same material culture and genetically nearly completely discontinuous with the previously resident population. Such discontinuity persists through to samples from the Bronze Age, documenting a demographic turnover at the onset of the Bronze Age that was crucial to understand the formation of the present-day British gene pool. The arrival of the BBC in Britain can thus be viewed as the western continuation of the massive movement of people that brought the Corded Ware Complex and steppe ancestry into central Europe a few hundred years before.

Ancient DNA has revolutionized our understanding of the history of the past. In a fundamental manner many archaeologists were wrong in assuming that the dominant dynamic of the spread of culture was that of the diffusion of ideas, as opposed to the movement of peoples. But to interpret these results it is clear that archaeological knowledge must be brought to bear, albeit updated with knew prior assumptions.

It would not be entirely surprising if the originators of a cultural complex transmitted it to another group, and then that culture “hitchhiked” on the demographic expansion of the receiving group. A good example would be Roman Catholic Christianity. The Iberians spread it to the New World, along with substantial demographic movement. But the religion itself did not spread to Iberia through migration, but rather cultural shift.