Did agriculture move in the hands of people, or with the words of people? Or, somewhat more prosaically:
Two alternative models have been proposed to explain the spread of agriculture in Europe during the Neolithic period. The demic diffusion model postulates the spreading of farmers from the Middle East along a Southeast to Northeast axis. Conversely, the cultural diffusion model assumes transmission of agricultural techniques without substantial movements of people.
Actually not just Europe. And the jury is still out. Two recent paper both tilt towards cultural diffusion, both in Europe (which is where the above quote comes from; but not everybody agrees) and Island SE Asia. This sort of work is mainly done by anthropologists and human geneticists. Sometimes the genetics of livestock or crops are brought into play, but only rarely both at the same time. A grand synthesis of human, livestock and crop genetic data, archaeology and anthropology remains to be done…