Migration and stratification are increasingly intertwined. One day soon it will be impossible to understand one without the other. Both focus on life chances. Stratification is about differential life chances - who gets what and why - and migration is about improving life chances - getting more of the good things of life. To examine the interconnections of migration and stratification, we address a mix of old and new questions, carrying out analyses newly enabled by a unique new data set on recent legal immigrants to the United States (the New Immigrant Survey). We look at immigrant processing and lost documents, depression due to the visa process, presentation of self, the race-ethnic composition of an immigrant cohort (made possible by the data for the first time since 1961), black immigration from Africa and the Americas, skin-color diversity among couples formed by U.S. citizen sponsors and immigrant spouses, and English fluency among children age 8-12 and their immigrant parents. We find, inter alia, that children of previously illegal parents are especially more likely to be fluent in English, that native-born U.S. citizen women tend to marry darker, that immigrant applicants who go through the visa process while already in the United States are more likely to have their documents lost and to suffer visa depression, and that immigration, by introducing accomplished black immigrants from Africa (notably via the visa lottery), threatens to overturn racial and skin color associations with skill. Our analyses show the mutual embeddedness of migration and stratification in the unfolding of the immigrants' and their children's life chances and the impacts on the stratification structure of the United States.