Recently, some museums like the J. Paul Getty Museum and Boston's Museum of Fine Arts have tightened their policies on acquisition of artifacts and have even returned artifacts to their country of origin.
The Getty repatriated a 4th century BC inscribed tombstone and a 5th century BC marble relief (which I posted about here) to Greece. They also returned to Italy artifacts stolen from the Greek ruins of Selinunte.
Boston's Museum of Fine Arts (MFA) also repatriated a set of illicitly gained artifacts to Italy after signing an agreement with the Italian Ministry of Culture. The artifacts included an amphora that depicts the murder of Atreus and dates to the 4th century BCE and a 5th century BCE lekythos.
The agreement between the MFA and the Italian Ministry of Culture makes allowances for the loan of other significant works and the acquisitions, information, conservation and archaeological investigation. The Getty's announcement that it has tightened acquisitions policies, however, doesn't appear to have overly impressed upon the Italian government. The Getty hasn't made the policy retroactive, which would require that they "relinquish scores of ancient items from its galleries and storerooms." But another reason for Italy's reluctance to sign an agreement with the Getty is there is a current legal case in Italian Court that involves former Getty curator Marion True, accused of being knowingly complicit in the acquisition of looted artifacts and Italian authorities are demanding the return of some 52 items in the Getty's possession.
There is still some apparent ground to cover with regard to antiquities acquisitions past, present and future, but these do appear to be steps in the right direction.