Following on from the unresolved question of whether material about a given culture is 'part' of that culture, I'm now going to shift to asking a similar question about inspiration and influence.
Hypothesis #1: if x is 'part' of culture C, material inspired by x is also part of culture C.
Hypothesis #2: if x is 'part' of culture C, material inspired by x may weaken the position of x as part of culture C.
I don't think anyone actually holds #1 in so thoroughgoing a form – it entails, for example, the surprising conclusion that Othello, inspired by an Italian text, is part of Italian culture – but presumably we mean something like it by the term 'Americanisation': not only the import in large quantities of U.S. media, but also their subsequent influence on domestic creativity and cultural expectations, is held up by the use of the word as a form of cultural encroachment. At any rate, getting clear about what's wrong with #1 may help clarify what does make a thing part of a culture. After all, when asking what culture consists of I presumably should be taking an interest in the transmission and interaction of ideas; and in many cases in which transmission happens we describe it by saying that one cultural item inspires another, the creative act being seen to depend on a previous item or corpus as a stimulus or catalyst.
Inspiration is a causal process, of a sort: the thing that inspires doesn't do so through any kind of activity in its own right, of course, but it is necessary for the resulting event of inspiration to take place, and hence for any consequent creativity to occur; and we construe this process as a sort of intellectual impact, in which the source of inspiration (the Muse, even) as grammatical subject 'inspires' the creative as object. So if (as seems impeccably plausible on the face of it) the transmission and propagation of ideas is important to culture, and by 'inspiration' we pick out processes by which it happens, how could inspiration not contribute to determination of cultural parthood? Moreover, if we wish to incline ourselves more towards #2 and say that to draw on the styles and motifs of another culture counts as 'cultural appropriation', x's powers of inspiration spilling over into effects within other cultures (D, E...) and thereby weakening x's position as a distinctive part of its own originating culture C—then on what grounds will we be claiming that the inspired has appropriated the inspirer, rather than that the inspirational culture is expanding its territory? Do such judgments draw on any understanding of how processes of inspiration operate, or are they entirely a matter of how cultural material is put to practical use as an instrument of (political) power?
It's a complication that 'inspiration' incorporates a lot of subtle distinctions and forms: homage, parody and caricature, satirisation, fictionalisation and cameo, ode and epitaph, and so on. Inspiration may be manifested in more or less overt emulation: besides the sort of appropriation in the arts which Young delineates into 'style appropriation', 'content appropriation', and so on (p. 5ff), it's possible to hold up more general or abstract aspects of a culture as inspirational, such as when peoples alleged to live in harmony with nature are held up as examples for the West to learn from and emulate. Besides emulation and adoption, inspiration may involve seeking to surpass others' already inspiring achievements; or it may even involve being negatively inspired to avoid emulation—in the sense that a practice common in some other culture but considered distasteful in our own may reinforce our opposition by giving us a concrete example of what we avoid. Inspiration may not even be consciously recognised by those undergoing it.
At a minimum, what I envisage for cultural 'inspiration' is that (1) experience of (some aspect of) C was a necessary condition for the production of x in the form it takes, and (2) the aspects of x under consideration were not simply copied from C. (Prototypes don't inspire production lines, and bootlegs are not products of inspiration. Nevertheless, I wonder whether I have worked this stipulation out far enough: it would seem natural enough, for example, to say that someone inspired me to emulate him in order to become like him.)
Ordinarily we might be inclined to say that a thing belongs to the culture to which its makers subscribe; but insofar as we can perceive elements of the trappings of some other culture in its composition, we may wonder whether the makers of the new work can in fact unreservedly be called its creators. This boils down to a question about the nature of creativity: if we can perceive the influence of x, which is part of culture C, in some work y produced by someone who is an outsider relative to C, are we to say that x-in-y is still part of C, and therefore that at least part of y is part of it; or are we instead to conclude that the alchemy of creativity can render x, though still recognisable as inspiration in y, not actually present in y?
On general principles of economy (and a disinclination to become embroiled in concerns about 'the imperialist claims of the Romantic author' (Coombe, 'The Properties of Culture and the Possession of Identity: Postcolonial Struggle and the Legal Imagination', in Borrowed Power)) I should prefer to avoid depending on any contested understanding of the nature of creativity, but this looks like a difficult thing to achieve. Culture is full of created things, and the question of who can claim credit for creating them naturally tends to loom large in patrimony disputes, making it hard to evade the question of what exactly creativity involves.
The question of method I then face is that of whether it's possible to talk about the role processes of inspiration might have in determining which cultures count the products of those processes as their parts, without making substantive assumptions about the nature of creativity in the process. If we introduce it as a reasonable-looking premise that the product of a creative act is by default part of the culture its creator subscribes to, then a fortiori a genuinely creative use of appropriated material, as opposed to a purely imitative one, will create something that qualifies as a part of the immediate creator's culture; the complaint about 'Americanisation' is more about perceived slavish imitation and importation than about creatively inspired adaptation. What isn't so easy is to establish the conditions in which a creation can be inspired by a part of another culture without incorporating it, since formal similarities alone aren't a sufficient indicator of appropriation between cultures: complaints about appropriation depend on actual acts of emulation having occurred. If two cultures independently developed styles, stories, etc. which happened to be similar, it's difficult to imagine either getting very far with a complaint about the other's use thereof.
Although with some digging I could no doubt find out more about what other commentators have had to say about inspiration and the forms of cross-cultural creativity, I persist in thinking it would be unsafe to rely on assumptions about anyone's being right about the matter. It's tempting, given that I've previously been happy to emphasise the mutually reinforcing potential of links between works without worrying much about cultural boundaries, to go back to talking about 'clusters of ideas' without worrying about cultural parthood; but when appropriation debates tend to depend on questions of what belongs to which culture, they rather have to be defused rather than ignored. The best I could do at this stage, perhaps, is investigate whether facts of cultural parthood lend themselves to being morally relevant within the framework I've been developing for thinking about cultural 'value'; but I suspect that a negative answer, on its own, would most likely invite criticism of the framework.