The Dead Sea scrolls were originally Jewish in origin, just as Jerusalem was a predominantly Jewish city. I would say that the the Israelis have a greater claim than the Jordanians.
I think "ownership" of things produced by people now long ago gone is ambiguous, and as long as it's ambiguous, it's just best that they be in good hands.
The Center for the Study of the Great Plains is the crown jewel of Yale University in Connecticut; I have no idea how that happened, but it happened.
Yale has more Nebraska history than all of the museums, libraries, and archives here in Nebraska.
Which is fine with us; Nebraska is a small state with limited resources, and taking care of these things would impose an onerous burden; and it is possible that Yale takes better care of these things than we would.
Yale does not forbid people from Nebraska from examining our history stored there, any more than the British forbid Greek tourists from visiting the Elgin Marbles.