No. Mystics are not required to believe in reincarnation or anything else unless they can prove it for themselves. Mystics are supposed to be independent thinkers and any views they may have on an afterlife (or prior-life) are their own. Mystics, once they’ve been at the practice for a while, do share some core experiences regarding contact with Universal Intelligence or Divinity. But reincarnation is not a core belief. Even people who hold the view of prior or future lives differ greatly on what such life signifies and how the process may occur. Reincarnation has been explained by different cultures in different times in a variety of ways. There has never been a definitive evaluation of the experience of reincarnation or rebirth – Buddhists, Hindus, Christians, Jews, Muslims, Gnostics, Native Americans, Theosophists, Pythagoreans, Wiccans, and all other spiritual groups that have mentioned the possibility of multiple lives at one time or another all differ on the specifics of the concept. So while it might be true that most people in the world hold some form of belief in reincarnation, no standard version of the theory exists.
In this experience, mystics are no different from the rest of the world. Some believe they have memories of a past life and some do not. Can this be explained? Not fully – there is no objective, verifiable evidence to support reincarnation, yet people who have had subjective experience with the phenomenon are adamant that reincarnation is a reality.