I assumed that they were relevant to the Judaic history of Mesopotamia, but yours is a good question...
On the other hand the western world does not readily entertain questions of original ownership or otherwise the museums in Europe would be half empty ;-)
The question of original ownership and how Iraq came into possession of these manuscripts is a good one, only because if the items were originally taken by either force, coercion, or theft then the moral outrage of having these stolen during the invasion of 2003 and then returned to the people of the original authors holds little over which to sympathize with Iraq.
If however it came into Iraq’s possession as a bequest from the previous owners, and its ownership chain is untainted by an immoral acquisition, then Iraq can honestly say that custodianship should revert back to them... However, I am inclined to think that if the Iraqi custodianship is going to be less than beneficial to the artifact, then temporary custodianship should go to the Israelis on the basis that they do not restrict access to the artifacts (reasonably) , and that the artifacts be actively studied and those studies published, until such time as Iraq can guarantee the safety and accessibility of the artifacts.