Research on scientific careers finds a mover's advantage. International migration correlates with increased visibility and productivity. However, if scientists who move internationally, on average, enter into more prestigious employments than they came from, extant research may overestimate the direct performance gains associated with international moves. Building on insights from the sociology of science and studies of international researcher mobility, we examine how changes in employment prestige shape international movers' performance returns to mobility. We follow a cohort of 167,014 European scientists to identify individuals that move to the USA and pair these migrants to non-mobile scientists with identical home institutions, research fields, and genders, giving a final sample of 3978 researchers. Using a difference-in-differences design, we show a substantial increase in the publishing rates and scientific impact of transatlantic migrants, compared to non-mobile scientists. However, most of the movers' mobility-related boost in citation and journal impact is attributable to changes in employment prestige. In contrast, we find limited effects of employment prestige on changes in migrants' publication rates. Overall, our study suggests large variations in the outcomes of transatlantic migration and reaffirms the citation-related "visibility advantage" tied to prestigious institutional locations.