This is a refined/sheafified version of this previos question of mine.

Let $(X,\mathcal{O}_X)$ be a ringed space or more in general a ringed stack, where the structure sheaf $\mathcal{O}_X$ is a sheaf of $\mathbb{K}$-algebras for some field $\mathbb{K}$. Then a global section of $\mathcal{O}_X$ can be thought of as a "scalar field" on $X$. In the "categorical progression" of higher $\mathbb{K}$-vector spaces, the field $\mathbb{K}$ is the 0th level. 1st level are classical $\mathbb{K}$-vector spaces, and so the "1st level version" of a section of $\mathcal{O}_X$ is (at least roughtly) a field of $\mathbb{K}$-vector spaces on $X$. A natural formalization of this rough idea is that of going from the sheaf $\mathcal{O}_X$ to the stack $\mathcal{O}_X$-Mod of sheaves of $\mathcal{O}_X$-modules on $X$ (maybe with suitable assumptions, e.g., one could be considering only quasi-coherent sheaves of $\mathcal{O}_X$-modules). A remarkable property of $\mathcal{O}_X$-modules is that they can be pushed forward along morphisms of ringed spaces. Now, the structure sheaf $\mathcal{O}_X$ we started with is in particular an $\mathcal{O}_X$-module, and pushing it forward along the terminal morphism $\pi:X\to \{pt\}$ we precisely get sections of $\mathcal{O}_X$ mentioned above.

Now we can make a further step, and go from the stack of $\mathcal{O}_X$-modules to the 2-stack of $\mathcal{O}_X$-algebras (with $\mathcal{O}_X$-$\mathcal{O}_X$-bimodules as 1-morphisms and morphisms of bimodules as 2-morphisms). My question is: can $\mathcal{O}_X$-algebras be pushed forward along morphisms of ringed spaces? (under which hypothesis?). In particular, considering $\mathcal{O}_X$ as an $\mathcal{O}_X$-algebra, what is the $\mathbb{K}$-algebra $\pi_*\mathcal{O}_X$? (the prototypical conjectural example of this in my mind is the following: if $G$ is a finite group, then $\pi_*\mathcal{O}_{pt//G}$ is Morita equivalent to $\mathbb{K}[G]$, the group algebra of $G$).

In the above, an $\mathcal{O}_X$-algebra is to be thought as of a placeholder for its category of modules, so in a non-affine situation it is actually deceitful to reason in terms of algebras and a better description would be thinking of the "second step" as $\mathcal{O}_X$-linear categories, with additive functors and natural transformations of these. In this more general setting, the pushforward to a point of $\mathcal{O}_X$ would be the "pushforward of the 2-category of $\mathcal{O}_X$-linear categories" and this should be a $\mathbb{K}$-linear category, but not necessarily the category of representations of an algebra. In particular, by abstract nonsense I would expect this pushforward to be the category of $\mathcal{O}_X$-modules.

Note that going one step backwards instead of one step forward, the existence of a pushforward is nontrivial: the pushforward of a section of $\mathcal{O}_X$ along $X\to \{pt\}$ is an integration of fuctions on $X$, so it requires additional data to be defined (a "measure").