Does the emigration of skilled individuals necessarily result in losses for source countries due to a brain drain? Combining industry-level patenting and migration data from 32 European countries, we show that emigration positively contributes to innovation in source countries and does not increase asymmetries in innovation levels between more and less advanced countries. We use changes in the labour mobility legislation within Europe as exogenous variation to establish causality. In addition, by analysing patent citation data, we provide evidence that these positive effects are driven by knowledge flows triggered by migrants. While skilled migrants are not inventing in their home country anymore, they contribute to cross-border knowledge and technology diffusion and thus help less advanced countries to catch up to the technology frontier.