A possible link to royalty and the Tunstalls is by the marriage of Thomas Tunstall, second son of Sir Thomas Tunstall and Eleanor FitzHugh. In the Visitation of Yorkshire, it gives his wife as the daughter of Bishop Neville. There were two Bishop Nevilles. The elder one was Bishop Robert de Neville, son of Ralph de Neville (Earl of Westmoreland and Joan Beaufort,daughter of John of Gaunt, son of Edward III. The other one was George de Neville, son of Richard de Neville (Earl of Salibury) and Alice de Montagu. George is the grandson of Ralph de Neville and Joan Beaufort, so either way there are descents from Edward III.
There could be problems with the descent for some of the children in that it appears that Thomas Tunstall may have been married twice, once to the daughter of Bishop Neville and the other possibly to a Conyers daughter. His son Culthbert Tunstall was also a Bishop. He was famous in that he was involved with Henry VIII and his divorces. He was the last Catholic Bishop of Durham. One article states that son Sir Brian Tunstall, the "shining Knight" immortalized by Sir Walter Scott is said to be a half brother to Bishop Culthbert Tunstall. Culthbert Tunstall was said to have been illigetimate and later made legitimate by the marriage of his mother and father. The Visitations of Yorkshire gives all the children from the Daughter of Bishop Neville.
An interesting note is that Bishop Robert de Neville had a brother named Culthbert. The name is not unusual in Yorkshire however.
There are two unidentified daughters in the Visitation of Yorks for Sir Thomas Tunstall and Eleanor FitzHugh. They give Elizabeth married to Sir Simon Norwich and Johanne married to Sir Roger Ward and Sir William Stapleton. But they do not name the other two daughter.
I am trying to place a daughter of FitzHugh that was married to Thomas Layton of Dalemaine, Cumberland.