The key to making people see the many flaws in their logic is to backtrack. A typical conversation might go as follows:
Me: Why is multiculturalism a good thing? Why is diversity our strength?
Multicultist: People from different backgrounds bring new ideas to the table. Ethnic food is amazing, and an all-white society would be boring.
Me: I agree (not really, but they don't know that yet!). I'd like to see human diversity preserved, on both the genetic and cultural levels.
Multicultist: What do you mean?
Me: Well, the end result of localized diversity is the eradication of global diversity.
Multicultist: How is that?
Me: Groups of people who live in close proximity to each other inevitably mix, especially when the cultural ethos of the host nation encourages it. The only possible result of this is that the host population becomes subsumed, genetically and culturally, in a sea of others, thus reducing global human diversity.
Multicultist: That's OK. Once we're all mixed, there will be an end to racism and the world will be great.
Me: *pretends to be confused and deep in thought* But didn't you just tell me how great an asset it was to have people from such diverse backgrounds? If everyone's background is the same, then we'll have lost that advantage forever. (Depending on the person's intelligence and interest levels, you can go on to talk about how genetic diversity on a global scale can help humanity survive epidemics, etc. etc.)
Multicultist: Wow, I hadn't thought of it that way.
From there, you can take it one of two ways. If they seem receptive to un-PC talk about genetics and human populations, you can strengthen the point by delving into genetic differences in intelligence, crime rate and per capita wealth by race/nation, etc. If not, just re-emphasize the point that immigration, integration, and mixing serve only to destroy human diversity.
Sorry for the lengthy ramble!