For the American side of my family, the real social mobility came in my grandparents' generation. My two American grandparents were born in 1911 and 1917, respectively. My grandfather was born at home on a homestead to parents who had very little education. His parents farmed. My grandmother's parents were also not educated. Her father worked for a railroad and later farmed.
In both cases, my great-grandparents were able to work hard enough that my grandparents were able to go to college. After WWII the GI Bill sent millions of Americans to college and my grandfather got a job as a professor, teaching them. As was the way in those days, my grandmother never had a full-time paying job, but did end up doing things like being on the school board in her town.
In this way, my grandparents were able to move from very humble beginnings to a place solidly within the middle class. Their achievements allowed my parents' generation and my own to remain in that same general socioeconomic status.