I don't think there is only one good solution for this. This is a very complex problem, which, in my opinion, needs diverse approaches and combined co-operations of the governments, museums, and international and intergovernmental institutions (like ICOM, for example) and agencies (like UNESCO) and organizations (like UNIDROIT).

There are already a couple of conventions (some of them I have posted in the comment below), which proved to be ineffective in the face of Iraqi (destroyed sites of Nineveh, looted Mosul museum and Mari temple, etc.) and Syrian(destroyed Palmyra) wars and not only (forever lost Buddhas of Bamiyan in Afghanistan). Those international conventions will never be 100% effective, unless all parties are involved and working on the problem and websites like Ebay and the auction houses have to be part of it.

What would be of a great help in this whole story, is if the cyber security specialists (please correct me if I used the wrong term for it)would co-operate and help track down the black markets.

I'm not a specialist myself and I believe there are far more experienced people working on this problem already. I just think that a combination of 'powers' would bring far more effective results.