There was no "old country" in my family as such. The newest arrival to the US that I know of would be one grandfather's parents, although he was born in the US. Both grandmothers were born into families that, as far as I know, had been in the US some time.
I can't really speak for what happened in the past with my ancestors. But in general, I think there was pressure to "become American". And I also assume that the changes of the 20th century had influence, as the world shifted from "old" ways to "new" ways. Both grandmothers, for example, probably started out using a wringer washer, but saw the dawn of automatics.
And there were probably some losses going into my parents' generation. I guess I could say my maternal grandmother lived in a certain small town America "culture." Meals were meat and potato, nothing fancy, and absolutely NOTHING foreign. Clothes were dried outside in summer. And everyone used the back door of the house--the front door was "saved for good."
My mother jettisoned a lot of the above. She used a dryer year round. Her cooking included ethnic dishes, and vegetarian dishes. And the back door was only used to go to the backyard, usually to take the dog out. How much of the above changes were my mother rebelling? Who knows? It was probably a mix of things. Some rebellion. Some "this method works better than the method I grew up with." Some my father. (Apparently, my mother made one small town dish for my father, once, and he took one look, and asked something like: "What the hell is this?") Some were undoubtedly practical. Why go to the back door when there is much easier to reach front door, positioned mere yards from the driveway and garage?
One "old America" touch that survived...for a while to my memory was having a vegetable garden. We had one in all but one house I grew up in (that one house was only a 1 year "get established in a new area" rental). Both grandparents had a garden (and both kept it running as long as possible). But it eventually went away in my parents generation. I can't remember if it happened before my family fell apart, or not. But certainly after my father left, the garden went idle. My mother wasn't interested in keeping it going--or interested enough.