You're most likely to come across the word transmigrate in a religious context. Your soul, some people believe, will transmigrate into a new body after death.
Another way to say transmigrate is reincarnate, or be born again after death. Many religions believe that people transmigrate after they die, including Buddhism and Hinduism as well as some Native American beliefs. Rarely, the verb transmigrate is used to mean "move to a new country," although migrate is much more common. The root is the Latin word transmigrat, "move from one place to another," from the prefix trans, "across or beyond," and migratus, "to move."