Will we all look Asian in the future?

What we look like in the future hinges on how integrated we are and whether or not a specific phenotype is consistently selected for by the entire world culture. The amount of variation present in the current human genome is considerable and will not be easily modified into a uniform "look" without very concerted effort. Frankly, the easiest way to achieve a uniform appearance in a world population would be genetic engineering (eugenics) and sterilization or euthanasia of all those nasty outliers in a population. Or simple destruction of populations and cultures that will contribute to continued variation, but that seems to be a bit negative in a world where the supposed reason we're all part of a world culture is that we all get along. So, the reality is, lots of variation will probably be accepted and, with no selection for specific traits the overall population will shift somewhat but the frequency of all the alleles will not change appreciably.