Here we go...Judy said.

|Now, she said that something to the effect that immigrants deserve full participation in America, and that this entails the right to change the |culture. This is precisely where things become fraught. Again, what is "integration"? How much integration do we have a right to expect? What |kinds of changes become intolerable?

I don't think it's a matter of deserving full participation, integration or righteous expectations. Integration is being free without the right to harm others - especially vulnerable co-residents. If taxes are paid... I'm good with immigrants trying to change our culture. No one controls culture for long.

Tim is making a good point. Culture is not a constant. Sometimes, people like to regard it as "consistent" by saying that society prefers certain "values," but even values are just social claims that serve a prescriptive forward-looking function or serve as means of characterizing retrospective judgments on past social behavior. I would argue that the social values of our country have changed throughout the years as public opinion has changed.
I had Sarah as a professor last fall for a class in which she touched briefly upon immigration and membership. I remember one point made was that if we apply an approach like John Rawls' "original position" to a broader global view that embodies immigration, the effect of immigration on a country's culture should not be a relevant moral consideration unless there's a real threat to liberal democratic values. I am not sure that there are any fundamental cultural aspects embedded in a sovereign state except for the fundamental social contract between its citizens and the state in the constitution. From this perspective, it would seem that integration is acceptable as long as the new immigrants abide by the conditions of the social contract.
Places of birth and the citizenship that it confers are arbitrary to the individual. A citizen's claim that they have any superior entitlement right to exclude immigrants seems relatively unfounded to me personally. It's also noteworthy to keep in mind that the United States had a relatively open immigration policy until the 1920s (with the exception of the Chinese Exclusion Act), which marked the emergence of the use of passports and the Immigration Act of 1924 that limited the inflow of immigrants.
Pressuring new immigrants to culturally assimilate evokes senses of cultural superiority, nearing senses of moral superiority that are relatively arbitrary. Cultural assimilation serves as erasure by communicating which cultures are more socially acceptable, whereas integration acknowledges each individual's right to autonomy without fallaciously pitting individual's (cultural) identities as "foreign" and against the state. Instead, a less xenophobic approach would be attempting to address these social conflicts from the context that these individuals are already part of the state. The argument that certain immigrants import "backwards" or morally depraved cultures is generalized and assumes that immigrants cannot or are not willing to abide by the terms of the social contract of a country which they are choosing to immigrate to, the act of which, by all means indicates that they demonstrate their acknowledgment of their intention to be just as legally bound to these rules as any other citizen in the country's jurisdiction. But, this point also brings up the question that if what we are really looking for is a community of not humans, who are fallible by nature and not by race, but a community of morally acceptable citizens, how is it that we choose not to deport natural born citizens who are just as much arbitrarily pre-dispositioned to be morally fallible? Our country unfairly endows additional rights and protections upon individuals born here than those who were not. I think this shows some of the conflicts we face when culture is seen as in conflict or inhibitive to civic participation.

This is pretty far after-the-fact. I too was uncertain about the usage of integration. That term is charged from the get-go. Assimilation is a softer way of talking about people fitting in. Dr. Song's points are well-taken by myself (and probably others). But, her explanation, I am certain, missed the mark for sensitive folks who have seen negative outcome, no matter which term is applied. Whether we are talking about swords ( double-edged):; razors (thin and sharp): or brooms ( tending to miss important minutiae), things get missed, if not lost, in translation. Some people, otherwise fit and willing to do so, have no wish to either assimilate or integrate. They simply desire the freedoms for which a free society claims to stand; the opportunities unavailable elsewhere.