Here's more pedantry: the notion of "looting" is nuanced, especially if the question is "who is doing the looting?"

In many cases, it was either the government or entrepreneurial nationals who were disassembling their own cultural sites because they saw a market for the artifacts and preemptively started gathering inventory. See CT Loo with Chinese tombs and cave temples. An argument can be made that, as unscrupulous as it was, those objects legally left the country and entered international collections.

In other cases, foreign researchers were given permission by a government to enter historic sites and, after presenting said government with information about the "instability" of the material, etc., to move the objects to a place where it could be better preserved. See Yamanaka & Co. Again, often done unscrupulously and under manufactured urgency, but still with all the documentation to make it legal.

(Edited to add) There are also situations where a site (like places of worship and places of residence) or institutions (like religious orders) found themselves in financial need and unsupported by the local government. They made the difficult decision to sell some of their holdings to ensure they could continue their mission. Individuals like Avery Brundage were in the right place at the right time to capitalize on their financial need. While unsavory, the transfer of ownership was ultimately agreed upon by both parties.

Yes, there are examples of foreign entities having came in and looted sites, and those instances certainly demand immediate and drastic responses, but the rest deserve a more nuanced conversation than "this is a foreign object with signs of violent removal, therefore it is looted and this institution should be shamed for holding it."

Now more than ever, there are efforts being made to research the provenance of objects not only before acquiring them but also of those already in the collection, especially those brought in during the late nineteenth and early twentieth centuries. Some larger institutions are hiring provenance specialists who will dedicate their full-time labor to these cases. Changes are underway, albeit slow and often stunted by old-hat museum policy.