I think a lot of it hinges on language.
Even though my children are growing up in their father's culture, he didn't want to share his dialect. He's more comfortable in French (the community language although Alsace is kind of bilingual). He grew up bilingually but all his education was in French.
We did put them in a German bilingual program but they really don't relate to being Alsatian. They refuse to put on the clothes and laugh at the music. I figure they need to get a bit more mature. Since I'm not Alsatian (although my family came from only a few miles away in Germany 4 generations ago by coincidence), I don't feel a great need to share my dh's culture with them. I'll point it out but that's about it. So this can happen even in the country of origin!
I know families who are really keen to do everything (music, dance, etc.) BUT the language. The kids don't really feel attached unless they actually speak it. They kind of "that's nice" and move on, like mine do with Alsatian culture.
But I always speak to them in English and encouraged them to reply in it. Only my son tried to reply in French but I nipped that in the bud. I'm so surprised not just that they're fluent in English but how well they relate to American culture. It's kind of their own version. When we go to the States, people often don't even realize that we live in France (until they speak to each other, which is always in French). They're quite proud and really gravitate to anything and everything American. It might help that I'm from California so people know about it, where it is, about the beaches, movies and Golden Gate Bridge. It's funny because they see NO conflict in being both French and American.
Now that they're older, I'm so glad that I kept it up. We can talk about personal and complex things in English and really make them understand. I've lived in France for 14 years and spoke French before moving here but I could never have a really close relationship with my children in anything but English (which my dh doesn't speak so that doesn't extend to adults).
I really urge you NOT to use the community language with your own children in public. This can send the message that your culture is something to hide or be ashamed of. Be proud (and no, it's not rude, don't blabber on in front of people-easy!)
I wondered how much religion would play a role. While we're a minority in both countries, my dh and I are of the same faith. That doesn't seem to be as crucial. It's even confusing since even the same religion can be practiced very differently. Perhaps it's made worse because my dh and I look so different, they have trouble linking us to the same background lol! So the one thing we had in common didn't really connect as well as I thought it would. They're proud of it but sometimes balk at the religion classes. They identify with it but they don't relate to very religious people that well (because we're NOT).
So language seems to be the key to avoiding disengagement, not religion, not music, not dance, etc. The bottom line is that if they don't have the language, they will never really attach themselves to the culture. We succeeded with the American and failed on the Alsatian. They might like the culture but really identifying with it is a different matter.