Michael Kremer, a Harvard economics professor with a track record of inventive ideas, and Tom Wilkening, a graduate student at MIT, published a possible solution earlier this year. Instead of flatly banning the export of antiquities, why not ban their sale but allow them to be rented?
The idea has a simple economic justification. Imagine a Malian sculpture, which is currently worth more to the British Museum than to the government of Mali. But it is possible that Mali will be much wealthier in a few decades than it is today and at that point will want the sculpture back. One of the easiest ways to arrange that pattern of possession is for Mali to lease the object to the British Museum for a few decades.
Beyond that smoothly plausible piece of textbook economics, the messy details also point in favor of leasing arrangements. If a poor country protects its antiquities with a blanket export ban, the government has to find cash to dig up, catalog, and store the things. Sensible governments would have other spending priorities, but that then leaves the artifacts in the ground, where they are difficult to protect from smugglers. A leasing arrangement would mean that an impoverished government could invite the Metropolitan Museum or the British Museum to go to all the trouble of excavating, researching, and protecting the treasures in exchange for, say, the right to exhibit them for 25 years.
... plausible in terms of economics, perhaps, but utterly implausible in terms of reality. While the focus on antiquities smuggling and the like has been focused on museums for the past while (for obvious reasons), this seems to ignore the fact that most of those museums received items from this or that private collection. Near as I can tell from auction coverage, the vast majority of stuff goes to private collections and isn't even available to the public. I really can't imagine the current 'pipeline' being affected at all by the possibility of 'renting' ... private collectors will just be able to buy directly from smugglers and middlemen. There's a lot more thought needed here.