National cultures and cultural artifacts are part of the human culture as a whole, and as such they belong to mankind.
There's hardly any good justification to destroy pieces of human culture, and misplaced nationalism definitely is not of them.
There's a certain phrase that people from my county use to express surprise and approval. It isn't widely known. When I use it around neighbors, they know what I'm conveying, and it identifies us as kindred.
I could tell you what it is, and you might like it. You might even start using it just because you like the sound of it. And it might become very popular, and soon, all kinds of people from all kinds of places might bond over its use. Hell, people might even make T-shirts.
But I choose not to. It's a shibboleth for me and the folks back home, and that use is valuable to us. You don't have the right to force me to share it.
If I wanted to explore its etymology, survey exactly how widely it's used, or argue that it's related to a similar phrase in a nearby region, I could certainly do those things and might choose to. But if I don't, human culture as a whole doesn't get a crack at it just because I passed on the opportunity.
It's a bit of a contrived example, because I'm not talking about the Bhagavad Gita here, but I hope you see what I'm getting at. The transmission of culture between peoples should be based on a mutual interest in sharing and learning, not some sense of global entitlement.
Assuming you're USAmerican, USA has "appropriated" the scientific culture of Russia by also going to space; appropriated the beer making culture of Europe; appropriated the Television/Computer culture of Britain; the number system born from Indian/Persian culture; appropriated the high-fashion culture of Paris/Milan/etc. in some of it's cities; appropriated the food culture of Mexico/Germany/Italy/wherever; appropriated the traditional medicines that gave rise to drugs like aspirin; et cetera.
Why do you want to stop positive advancements from being adopted by everyone, is it just greed? Why do you care what culture the music someone else listens to was born from.
Of course you should be allowed to not adopt other's culture if that's your choice, only use anything you've created from scratch yourself (tool use I think arose out of E. African culture, better avoid that) - otherwise you're appropriating somebody's culture in some way.
"The transmission of culture between peoples should be based on a mutual interest in sharing and learning, not some sense of global entitlement."
If something in your culture benefits me, why does my use of it have to be dependent on that also benefiting you? Indeed if its a loss for a small people group but a benefit globally then surely that's a positive thing.
Would you for example argue lack of distinctiveness as a negative thing, in which case you could be blocking all sharing by demanding "mutual interest".
Though I'm interested if you can give specifics, like tell us what culture you identify with primarily and what aspects of that culture you want to keep for only people you identify as 'in' (belonging, if you will).
Maybe expand and show what you're foregoing from other cultures, like if you're not Chinese so avoid cultivated rice, or whatever.
But I certainly don't want a bunch of carpetbaggers helping themselves to my culture in a way that would erode its meaning. So as a living member of the culture, I choose to share or not share based on what I value, why I value it, and how best I judge to nurture it. Would it be better for me, my people, and humanity if I shared this? Do I judge there is more value in not sharing while we are alive? Do I have a responsibility to preserve and transmit my culture--at least among my people?
Simple language is an example that breaks down fast, because I don't really lose anything by teaching you a phrase and its meaning. If my words live in one more mouth, there aren't fewer words for me to say. But even the meaning of words can change based on who uses them and how.
A better example might be a sacred text. I may have a "family Bible" handed down through centuries. The book itself might be a valuable cultural artifact, and the individual contributions by my ancestors might constitute a body of knowledge of interest to others. But it isn't anyone else's right to demand access to my "family Bible." There is no onus on my family to produce a digital copy, no moral obligation to permit visitors to leaf through its pages.
And you certainly may not hire men with guns to come into my home and take it just because I leave it on the coffee table instead of a safe. You may not kick in my door to preserve it just because the cat sleeps on it. But when I am dead and no kin remain to claim it? Take it, study it, put it in a museum.
But archaeological remains aren't really necessarily an artifact of the culture that lives there now. Sometimes there is a continuous cultural tradition that claims them, and sometimes not. The Mississippian mounds of the midwestern US aren't an artifact of the U.S. of A., for example—they were in danger of being looted by Americans more than anything, in the 19th century certainly.
And much as an international militia for the defense of archaeological remains is a bad idea, I think the OP was talking more about ISIS blowing up the Temple of Baalshamin in Palmyra or the Taliban blowing up Buddhist statues in Afghanistan. The locals are likely descended from the people who built those structures, but we do all lose something. These religious structures were not shibboleths, but public structures into which people poured their creativity, their personality, their values, their traditions to communicate them within and beyond their community. I am not a Christian, but I am moved by the Sistine Chapel—this was Michaelangelo's goal. I am moved by the Hagia Sophia, by the Ishtar Gate.
If the Romans decide to blow up the Vatican, or the Canadians invade Missouri and carry off the artifacts of Cahokia, it's a loss to more than the residents of Rome and St. Louis. Even if (as with Palmyra &c.) it's locals who object most strongly and are the most injured.
Sympathy is the appropriate response. Being ready to help people defend their culture in a conflict is fine, being ready to help people preserve culture before a conflict is fine, being available to help people recover what they can after a conflict is fine, but undermining people to preserve your expectation of value (like by hiring mercenaries to secure archives in a conflict zone) is not. That is an important difference.
The destruction of cultural artifacts previously unknown by the local mainstream is thus deplored, not because it had any real intrinsic value, but because it is no longer available for someone to show off and say, "look at this cool old thing that I found, that none of you culturally ignorant heathens have ever seen before."
As N-thousand-years-old artifacts are a scarce resource, whenever some hairy barbarian burns a book that no one cares about, he deprives some hipster of a critical opportunity to gain prestige in front of their friends for about 20 minutes.
Now, some people may become upset that elements of what they perceive as their own culture are essentially being used as toys and amusements for someone else's culture, but it is an essential element of culture that it cannot be controlled in such a way that such uses could be prevented. Culture is broadcast. You put something out there, and it is either adopted and rebroadcast, or it fades into obscurity. Once it is part of the culture, it isn't fully yours any more.
There is no protection for culture. If the consensus changes the meaning of something you liked to something you don't, you just have to move on to liking something else. If someone manages to make rompers for grown men fashionable, you just suck it up and wait patiently for the fad to die. You don't have to wear one, unless you want to participate in that part of the culture. And you don't have to rescue one from a thrift shop as a cultural artifact after it goes away. If you did, it would just be some thing to stick in a museum, so future visitors could see it and giggle, say "hmmmm", or vow "never again".
Not all cultural artifacts are worthy of "saving". For all we know, those unique manuscripts that might be put to the torch when the invaders march in are someone else's cultural equivalent of a Hitler Pepe meme or a man-romper silkscreened with Kim Jong-Il's face at 500% actual size.
Cultural traditions are exactly those things that generation after generation of a community has seen the value of preserving.
It would be acceptable to offer to buy the ones you intend to preserve, or to donate to a museum to preserve the ones they see as important. Archaeology at gunpoint isn't good for the artifacts, isn't good for the people who had them, and isn't good for the people that take them. The best you could do in that vein would be to create a new cultural artifact featuring an adventurer-archaeologist that pilfers the important artifact you would like to preserve as the main plot MacGuffin. And to not make it about stupid crystal-skulled aliens.
Everywhere else I have seen it used there has always been an implication that the making of distinctions like this was negative.
I'm not making a value judgement here, it just struck me as unusual.