George Borjas looks at social mobility across generations for immigrants to the U.S.:
Making it in America: Social Mobility in the Immigrant Population, by George J. Borjas, NBER WP 12088: Abstract The ultimate impact of immigration on the United States obviously depends not only on the economic, social, political, and cultural shifts that take place ..., but also on the adjustment process experienced by the immigrant household across generations. This paper documents the evidence on social mobility in the immigrant population and summarizes some of the lessons implied by the evidence. There is significant economic "catching up" between the first and second generations, with the relative wage of the second generation being, on average, about 5 to 10 percent higher than that of the first generation. At the same time, there is a strong positive correlation between the socioeconomic outcomes experienced by ethnic groups in the immigrant generation and the outcomes experienced by their children, and a weaker correlation between the immigrants and their grandchildren. In rough terms, about half of the differences in relative economic status across ethnic groups observed in one generation persist into the next. As a result, the very large ethnic differences in economic status that characterize the current immigrant population will likely dominate discussions of American social policy for much of the 21st century.