"Race" is a social construction and so who is "white" has changed across time and place. In the US, it is true that Southern and Eastern Europeans who were not considered white at one point, later became white. This process, and the role played by organized labor, is documented by Herbert Hill, ("Race and Ethnicity in Organized Labor: The Historical Sources of Resistance to Affirmative Action" JOURNAL OF INTERGROUP RELATIONS, Winter, 1984; "Black Labor and Affirmative Action: An Historical Perspective" in Shulman and Darity(eds): THE QUESTION OF DISCRIMINATION, 1989, Wesleyan University press), among others.
Hill outlines the way in which wave after wave of European immigrants were able to "acculturate", "assimilate," go through the "Americanization" (whitenization) process. Hill does not say that European immigrants had it easy, not at all. But they were eventually accepted. Many union leaders, for example Samuel Gompers, President of the AFL from 1886 to 1924, argued that Chinese, Japanese and African Americans were "unassimilable" and should therefore be excluded. One of Hill's most fascinating points is that racist ideology was part of the Americanization process. I can't do justice to Hill's work, but it is a disturbing history.