I wrote about this in this earlier reply to another answer. In it I'm drawing mostly from Katherine Cook's work, afaik still the only monograph on the topic. She analysed a large number of judicial/inquisition documents that show that there were many cases of moriscos crossing over to Spanish America illicitly - similar to Jewish conversos. In contrast, it seems that they held lower positions, and often tried to "blend in" racially, as the presence of moriscos in its overseas possessions was a major issue of legimacy for the Spanish Crown. Then again, there also cases of moriscos wanting to be recognized as Muslims in order to escape persecution through the Inquisition.