Tudors (Non-royal)

The marriage of Jasper Tudor and Catherine Woodville produce several surviving children. They would be related to the royal line but not royal themselves. Could they have succession rights?
 
They would have a minor claim to the French throne (if we ignore salic law) but none at all to England. However they would probably be well treated by subsequent Tudor monarchs.
 
One of them might provide a husband for Mary; that way it's still the Tudors even if Henry VIII never gets a male heir.
 
Is Jasper had a surviving male heir, he might be married also to Elizabeth. It is also possible that Henry VIII might try to get a ruling allowing the heir to inherit.

Maybe as some form of adoption?
 
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