They are stolen property that should be returned to their rightful owners, The Greek people
By that standard, I would be able to run the decendents of some Kansas Redlegs off a couple of farms that my g'g'grandad owned in Cass County Missouri and take possesion. Nice big farms, too. Of course, there are some folks with a little Osage blood who might have something to say about it.
Elgin stole 'em fair and square. They would have probably been burned for mortar by the locals if he had not done so. The current Greek government has no significant continuity with the people who made the art. If the Greeks want them, let them make an offer.
Er... Elgin saved them from almost certain destruction, picking thum up from the ruins of the parthenon. He did it with the permission of the Greek government, under the supervision on site of Greek officials. Under what law is the British Museum beholden to return them.
If I came to your house and found a painting you had thrown away in your trash and asked if I could have it and you said yes, would your great grandson have any legal right to demand it back from my family in 60 years?
If they were stolen, they should be returned to the Ottomans.