There's 2 separate Hague conventions establishing an international framework for how to protect heritage in conflict regions. They're not perfect (like pretty much any convention), but they address all the basic issues like sheltering artifacts abroad and dedicating military units to prevent destruction.

Also, western institutions have not been ideal stewards themselves, historically. The Pergamon kept the Ishtar gate through bombings in WW2 and the GDR. The British Museum has lost untold numbers of artifacts because they don't even have the resources to do a complete catalog of their collection, let alone properly conserve them.