The issue is complicated, because in order for multiple cultures to coexist, either one culture must have hegemony in some sense, or some new cultural medium has to be constructed as a compromise between the many. Multiple cultures with very different values cannot fully integrate.
So, let's take an example: Let's say that you're hiring employees. You have a culturally based expectation that employees are there on the dot. Your workers are from a culture that doesn't value punctuality. Whose values will win? How about this, your culture is willing to tolerate other religions or a certain racial/cultural sub-group, and their culture isn't. Your culture believes that women's education is a matter of moral regard and that treating women as less valuable than men is a violation of human dignity and even child abuse, and they don't care whether women are even literate.
When we have a conflict, the simple answer is that in the West, Western values dominate and incorporate more and more things into them. Western institutions control society and even increasingly try to incorporate other aspects into them.(think about Che Guevara shirts and how that's an effort of our current institutional framework to even incorporate dissonant values into itself) Other cultures are tolerated because the West values tolerance, but if a major conflict or an area where our tolerance runs thin occurs, the dominant cultural beliefs feel the importance of their role to step in. Western culture enforces LOTS of its mandates, all of the time. Even multiculturalism, in so much as it exists, is an outgrowth of Western culture, than it is a neutral thing that emerged out of some bargaining session with other cultures.
And I think that because it's a construct that acts as if it is an essence and deeply rooted, when in fact, it really can never be anything but Western, it becomes somewhat confusing.