It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ the canonical map

$$ \bigoplus_i H^*(X; \mathcal{F}_i) \to H^*(X; \bigoplus_i \mathcal{F}_i) $$

is an isomorphism.

My line of interest is whether there are other situations where these kinds of things are true. One example came to my mind:

First observe that for a finite group $G$ and a sequence of $G$-modules $M_i$ the canonical map

$$ \bigoplus_i H^*(G; M_i) \to H^*(X; \bigoplus_i M_i) $$

is an isomorphism. Indeed, the bar resolution of $\mathbb{Z}$ over $\mathbb{Z}[G]$ is finitely generated in every degree. Thus, homming out ouf this resolution commutes with direct sums, as does homology.

Now let $X$ be a scheme with an action by a finite group $G$. Dividing by $G$, we get a map $\pi: X \to X//G$. For every quasi-coherent sheaf $\mathcal{F}$, we have a spectral sequence

$$ H^p(G; H^q(X; \pi^*\mathcal{F})) \Rightarrow H^{p+q}(X//G; \mathcal{F}) $$

(at least if $X//G$ is flat over $\mathbb{Z}$).

It easily follow that under these conditions (and $X$ noetherian) for every collection of quasi-coherent sheaves $\mathcal{F}_i$ on $X//G$, we have an isomorphism

$$ \bigoplus_i H^*(X//G; \mathcal{F}_i) \to H^*(X//G; \bigoplus_i \mathcal{F}_i) $$

Having these examples is not totally satisfactory since there might be a more general theorem with a uniform proof. So, my question is:

What are reasonably general conditions on a Deligne--Mumford stack such that arbitrary direct sums of quasi-coherent sheaves commute with cohomology?