Plate tectonics shapes Earth's surface, and is linked to motions within its deep interior^1,2. Cold oceanic lithosphere sinks into the mantle, and hot mantle plumes rise from the deep Earth, leading to volcanism^3,4. Volcanic eruptions over the past 320 million years have been linked to two large structures at the base of the mantle presently under Africa and the Pacific Ocean^5,6. This has led to the hypothesis that these basal mantle structures have been stationary over geological time^7,8, in contrast to observations and models suggesting that tectonic plates^9,10, subduction zones^11-14 and mantle plumes^15,16 have been mobile, and that basal mantle structures are presently deforming^17,18. Here we reconstruct mantle flow from one billion years ago to the present day to show that the history of volcanism is statistically as consistent with mobile basal mantle structures as with fixed ones. In our reconstructions, cold lithosphere sank deep into the African hemisphere between 740 and 500 million years ago, and from 400 million years ago the structure beneath Africa progressively assembled, pushed by peri-Gondwana slabs, to become a coherent structure as recently as 60 million years ago. Our mantle flow models suggest that basal mantle structures are mobile, and aggregate and disperse over time, similarly to continents at Earth's surface⁹. Our models also predict the presence of continental material in the mantle beneath Africa, consistent with geochemical data^19,20.