Research on international migration and innovation relies heavily on inventor and patent data, with "migrant inventors" attracting a great deal of attention, especially for what concerns their role in easing the international transfer of knowledge. This hides the fact that many of them move to their host country before starting their inventive career or even before completing their education. We discuss the conceptual and practical difficulties that stand in the way of investigating other likely channels of influence of inventor's migration on innovation, namely the easing of skill shortages and the increase of variety in inventive teams, firms, and location.