Listen to this podcast and write a brief summary of what happened to these ethnic Southern and Eastern Europeans. https://beta.prx.org/stories/23738

The groups being discussed as "ethnic Europeans" begins with the Irish, used as a base of comparison to evaluate ethnic Europeans who came to America later, in the 20th century. The Irish assimilated into American white culture, an assimilation that was boosted by developing a powerful East coast political machine. After the Irish, came the Italians, Greeks, Poles and Jews. Their skin and hair colors made them more distinguishable as "different" than the Irish hair and skin coloring, and their routes to assimilation into the culture of white English and Central European Protestants were made much harder by these distinguishing traits. These ethnic European groups are distinguished and set apart from "white" by dark, often curly, hair and "olive" skin.  
Using the Greek experience as an illustration of the experiences these groups shared, The Color Initiative podcast describes the typical racially punitive white reaction to these groups. In Omaha, Nebraska, in 1909, every Greek home was razed by fire in an ethnic cleansing when a police officer was killed as the result of a Greek man fleeing after being accused of having an affair "with a white woman," which is typically a racial crime in the South. The dark curly hair and dark skin of the Greeks associated them, in the cultural mind's-eye, with African Americans and Mexicans. Organizations like the American Hellenic Education Progressive Society grew up in response to hate crimes perpetrate against these ethnic groups, which were seen--as for example in the American South and Nebraska--as comprising a separate race: e.g., the Ku Klux Klan perceived Greeks as comprising a separate race and attacked Greeks on racial grounds.
Greeks and other ethnic Europeans living in Chicago's inner city ghetto enclaves were dispersed when Greek Town and other ethnic "towns" were bulldozed to make way for redevelopment, such as for a university site in Chicago. These dispersed groups--still using Greeks as an illustration--moved to suburbs where returning ethnic soldiers, having fought alongside other ethnic groups and whites in foxholes, becoming united in the brotherhood of war, had already settled. Once in white neighborhoods, speaking Greek only privately in homes and speaking English and only English everywhere in public, Greeks came to be seen as white European Americans. 
In concluding remarks, the podcast compares these ethnic European immigrations to the immigrations of the 21st century. The 21st century immigration movements are coming from Asia, including Southeast Asia, India and Pakistan, Latin America and Africa. Not only do these groups bring more distinctively different skin and hair color (and type) with them, they bring religious practices, manners of dress, cultural mores and religious governance that are more extreme in their distinctiveness than what the Europeans brought, which was Catholicism, Judaism and Eastern Orthodoxy, with variations on European dress and manner. The prediction is that these new immigration groups will find it harder to assimilate because their cultures require that they exclude themselves from much of white culture and society, and they will find it harder to be assimilated and to be accepted as part of the traditional, white dominated, Protestant dominated American culture.
https://beta.prx.org/stories/23738