let me start by saying this isn't something i've thought about in a lot of detail, so this is just a sketch of some of my initial thoughts, which i'm posting in the hope of getting some feedback rather than to make an actual argument. so that said...
in a broad sense, 'cultural appropriation' seems to describe a real thing: members of a dominant or majority culture adopting cultural traditions or traits of a minority culture, while at the same time oppressing the culture they borrow from.
i don't have a problem with this general framing. it seems like something bad is going on here.
but i'm less convinced by some of the responses to this problem, which often seem rooted in liberal or leftist identity politics and lead to conclusions that seem incompatible with anarchy. two things in particular seem troubling:
firstly, because these leftist reactions are rooted in identity politics, they necessarily depend on someone's claim to an identity: if someone 'is black' or 'is native american' then they are permitted to make use of black culture or native american culture, while if they don't possess that identity then they aren't. but this creates the same problem found in other expressions of identity politics, which is that these identities are not (and should not be) clearly delineated borders, and trying to make them so means reifying identity in ways which are explicitly harmful, such as 'blood quantum' rules or the white supremacist conception of 'white' meaning 'anyone not tainted by non-white blood'. i find this incompatible with my anarchism, at least, because i cannot find any path to individual or social liberation through arbitrary division of people into named categories.
secondly, these restrictions on appropriation are often framed in terms of 'ownership' of ideas. for example, i recently leaned about the idea of 'indigenous intellectual property', an attempt to bestow legal ownership of cultural 'IP' on indigenous people. but even when not framed in an explicitly legal context like that, the rhetoric is often that certain aspects of culture are 'owned' by a particular group and therefore they should have the right to control how other people use those ideas. since 'rights' and 'ownership' (whether of physical or 'intellectual' property) are incompatible with anarchism, i don't think this framing can be supportable. but if that's true, then there's no basis for anyone to object (in a general sense) to 'appropriation' of culture.
all that aside, some of the things often identified as cultural appropriation are actually harmful, but perhaps these can be re-framed. for example, someone appropriating AAVE to make fun of black people is harmful, but the harm there is racism. someone appropriating native american culture to sell it for profit is harmful, but the harm there is from the economic system which allows them to do that.
is there any framing of cultural appropriation which is both defensible from an anarchist position, and can be usefully applied to create meaningful action beyond action anarchists would already naturally support?