Yes / Bell Beaker pottery spread across western and central Europe beginning around 2750 BCE before disappearing between 2200-1800 BCE. The mechanism of its expansion is a topic of long-standing debate, with support for both cultural diffusion and human migration. We present new genome-wide ancient DNA data from 170 Neolithic, Copper Age and Bronze Age Europeans, including 100 Beaker-associated individuals. In contrast to the Corded Ware Complex, which has previously been identified as arriving in central Europe following migration from the east, we observe limited genetic affinity between Iberian and central European Beaker Complex-associated individuals, and thus exclude migration as a significant mechanism of spread between these two regions. However, human migration did have an important role in the further dissemination of the Beaker Complex, which we document most clearly in Britain using data from 80 newly reported individuals dating to 3900-1200 BCE. British Neolithic farmers were genetically similar to contemporary populations in continental Europe and in particular to Neolithic Iberians, suggesting that a portion of the farmer ancestry in Britain came from the Mediterranean rather than the Danubian route of farming expansion. Beginning with the Beaker period, and continuing through the Bronze Age, all British individuals harboured high proportions of Steppe ancestry and were genetically closely related to Beaker-associated individuals from the Lower Rhine area. We use these observations to show that the spread of the Beaker Complex to Britain was mediated by migration from the continent that replaced >90% of Britain's Neolithic gene pool within a few hundred years, continuing the process that brought Steppe ancestry into central and northern Europe 400 years earlier.