Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + \frac{ \sum_{i=1}^k n_i}{2}$ for positive natural numbers $n_1,\dots, n_k$, then there are several natural maps

$$ \overline{\mathcal M}_{1,n_1} \times \dots \times \overline{\mathcal M}_{1, n_k} \to \overline{\mathcal M}_{g}$$

one for each perfect matching of the $\sum_{i=1}^k n_i$ marked points that connects the $k$ elliptic curves.

Is the functoriality map $H^i\left(\overline{\mathcal M}_{g}, \mathcal O_{\overline{\mathcal M}_{g}}\right) \to H^i\left( \overline{\mathcal M}_{1,n_1} \times \dots \times \overline{\mathcal M}_{1, n_k}, \mathcal O_{ \overline{\mathcal M}_{1,n_1} \times \dots \times \overline{\mathcal M}_{1, n_k}}\right)$ ever nontrivial for $i>0$?

The motivation is that this would imply that some piece of the motive of $\overline{\mathcal M}_g$ is easy to understand, because it will be a tensor product of the motives of modular forms. You may of course substitute "Hodge structure" or "Galois representation" in for "motive" if you prefer.

I think not because it seems like that would make it "too easy" to understand part of the cohomology, but maybe it's difficult just because the map is sometimes zero and sometimes nonzero and there's no good way of telling when.