How much do schools mediate the international transmission of income?

Not that much.  From Jesse Rothstein:

Chetty et al. (2014b) show that children from low-income families achieve higher adult incomes, relative to those from higher income families, in some commuting zones (CZs) than in others. I investigate whether children’s educational outcomes help to explain the between-CZ differences. I find little evidence that the quality of schools is a key mechanism driving variation in intergenerational mobility. While CZs with stronger intergenerational income transmission have somewhat stronger transmission of parental income to children’s educational attainment and achievement, on average, neither can explain a large share of the between-CZ variation. Marriage patterns explain two-fifths of the variation in income transmission, human capital accumulation and returns to human capital each explain only one-ninth, and the remainder of the variation (about one-third) reflects differences in earnings between children from high- and low-income families that are not mediated by human capital. This points to job networks and the structure of local labor and marriage markets, rather than the education system, as likely factors influencing intergenerational economic mobility.

Here is the NBER working paper.