We propose a novel perspective on migration and cultural change by asking both theoretically and empirically - and from a global viewpoint - whether migration is a source of cultural convergence or divergence between home and host countries. Our theoretical model derives distinctive testable predictions as to the sign and direction ofconvergence for various compositional and cultural diffusion mechanisms. We use the World Value Survey for 1981-2014 to build time-varying measures of cultural similarity for a large number of country pairs and exploit within country-pair variation over time. Our results support migration-based cultural convergence, with cultural remittances as its main driver. In other words and in contrast to the populist narrative, we find that while immigrants do act as vectors of cultural diffusion, this is mostly to export the host country culture back home.