He did not “invade”, but “returned from exile”. The direct line of the House of Lancaster had been wiped out during the “Wars of the Roses”, and, being indirectly descended from John of Gaunt, he was now the “Lancastrian champion”.
By the time that John of Gaunt had been able to marry Katherine Swynford legally though, their children were in their 20s, but when he “made an honest woman” of his long-time mistress, Parliament legalized their offspring, giving them the “Beaufort” title, and an estate in Gloucestershire - Badminton. But it also specified that they could NOT use their Princely ancestry to justify any claim the Throne.
Margaret Beaufort married Owen Tudor, a Welsh gentleman, and Henry was their son, but defeating and killing Richard III at Bosworth did NOT make him King - he had to call Parliament, and ask it to repeal the Act barring him from the succession. This it did, in return for a promise to end the long, internecine, conflicts, and to provide “good Government”. Which he did, very effectively, transforming English society from one of mediaeval anarchy into its “modern” form, based upon the “Rule of Law”.