I think one of the big barriers to getting to the bottom of the cultural (mis)appropriation question is the (conscious or unconscious) tendency for many people on both sides of the debate to frame it in terms of questions of property (specifically post-Enlightenment Western ideas of property). (See here
for a good, albeit inconclusive, discussion that mostly avoids getting trapped in the property metaphor.)
For example, cultural elements often framed as public goods as justification for saying appropriation is no problem -- there's not some finite amount of a culture's music (or whatever) to go around such that an outsider using it prevents an insider from doing so as well. This line of thinking depends on treating cultural elements as fixed quasi-objects. But that's not how cultural elements work. Instead, they are embedded social practices. Because they are practices, they are subject to structurational shifts -- the element is reproduced every time it is used, and may be reproduced in slightly different ways (think of how the sounds and meanings of words change over time). Because they are embedded, the significance of a cultural element is as much a matter of its relationship to other elements as it is the inherent properties of a discrete "thing" -- so placing it in another context, using it in a different way than the originators*, alters it. And because they are social, each person has to live with the consequences of how the cultural elements they practice are practiced by others. Thus the significance of the appropriated version of the element can drown out the original.
All of this applies intra-culturally as well as inter-culturally -- a person's use of "their own" culture is not innocent (assuming such boundaries can even be clearly drawn -- another drawback of the property metaphor is that it assumes that cultures can be treated as discrete and internally unified units like property-owning persons and corporations are). But the power differentials between people of different cultures create special responsibilities in adopting cultural elements which are already part of someone else's way of life. This is a responsibility not to practice that element in a way that harms the originators -- either directly (by becoming a part of a social system that disadvantages them, as when another culture's art is used to signify exoticness) or indirectly (by encroaching on and redefining the practice of it by the originators).
Thinking this way I think explains the two most common ways to protect against misappropriation -- "become authentic" and "acknowledge inauthenticity." A call to become authentic -- to fully understand the originator culture and practice the element in question only within the same context that the originators practice it. Doing so cuts down on the potentially harmful distortions of reproducing the element in a new context. The strategy of "acknowledge inauthenticity" goes the other way, seeking to essentially build a wall or even make the element as practiced by the appropriator a "different" element than the one practiced by the originators (which is treated as an "inspiration"). Done right, this can cut off the effects of the appropriator's practice from the originators', though it also cuts off any attempt by the appropriator to benefit from a perceived connection to the originators (e.g. by presenting oneself as an authentic purveyor of ancient wisdom). I don't think these are the only two options (nor are they failsafe). Perhaps thinking outside the property metaphor can help give guidance to attempts to walk a third route.
As a final note, I think another advantage of getting away from the property metaphor is that it allows us to look beyond the question of the blame or responsibility of the direct appropriator. For example, several commentators in the above-linked post point out that the harm in Amy Winehouse's appropriation of black soul music is as much the fault of the music industry, critics, and fans (who treat her as more acceptable than black soul musicians) as it is of Winehouse herself.
*The term "originators" is relative -- it refers to the people the maybe-appropriator got the element from, regardless of whether they made it up out of whole cloth or (more likely) adapted or appropriated it from an earlier group. The claims of the originators are not, in my view, primarily based on invention or legitimate acquisition (a la a Lockean or Nozickian conception of property), but rather on their numbers and the degree to which their way of life is invested in it, which therefore determines the amount of harm that would be done by misappropriation.