This paper shows that the wage assimilation of immigrants is the result of the intricate interplay between individual skill accumulation and dynamic labor market equilibrium effects. When immigrants and natives are imperfect substitutes, rising immigrant inflows widen the wage gap between them. Using a production function framework in which workers supply both general and host-country-specific skills, we show that this labor market competition channel explains about one fifth of the large increase in the average immigrant-native wage gap across arrival cohorts in the United States since the 1960s. This figure increases to one third after also accounting for relative demand shifts due to technological change.