Éowyn is actually part Dúnadan from her grandmother Morwen which is why she and her family are a bit different from the other Rohirrim. Her grandmother was related to Dol Amroth's Prince and she actually took after her grandmother in form.
With intermingling some genetic characteristics were expressed where others were not. This can be seen in the two brothers Boromir and Faramir. Both were Dúnedain, but in Faramir all or most of the genes of their people were expressed. The same even occured in Éomer and Éowyn, though hardly to such a degree.
However, Théoden who had a Dúnadan mother, when compared to Denethor is clearly not even in the same class as per Gandalf.
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Originally Posted by Minas Tirith
Denethor is of another sort, proud and subtle, a man of far greater lineage and power, though he is not called a king.
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Since the rebellion in Númenor and the actual destruction of that land, the Dúnedain have been diminishing as a people. Their lifespans have shortened, they are not as vigorous as they once were, and as I said generally not all of their genes found expression. The King's house used to live for at least 400 years of age, Aragorn lived to 210 years and all of the Chiefs in the North generally lived to about 160 years unless killed.
I'd say they would not lose much of their pedigree, but they would continue to diminish in lifespan as all the Dúnedain were. By the time of the end of the Third Age they generally outlived other Men like the Rohirrim only 20- 30 years.