Modern-day missionaries try to build up the local church where they are. Some people do end up wanting to emigrate to Utah, but this is not something the Church encourages.
Go back to the 19th century, and the LDS Church was heavily involved in encouraging people to convert to their religion and immigrate to Utah. Many of their missionaries in Europe were not just seeking converts, but also immigrants.
Nowadays, the LDS Church doesn't put any great emphasis on encouraging their overseas flock to immigrate to Utah. But if they changed their mind about that, I doubt LDS immigrants from Asia or Latin America would destroy the society of Mormon Utah, any more than 19th century European Mormon immigrants did. If anything, I think it would be less destructive than US intrastate migration of non-Mormons into Utah, which has been going on for a long time now.
I find your use of the term "gross" inexplicable. Are you positing that you can have a high trust, high cohesion, orderly society that accepts lots of immigrants? If so, show me one. Otherwise, your position seems to be that immigration is a moral requirement in and of itself, regardless of the consequences it has for societal order.