If $\mathscr{X}$ is a stack over some base ring $k$ (if you are not familiar with stacks, read "schemes" here), we may consider it as a pseudofunctor $\mathscr{X} : \mathsf{CAlg}(k) \to \mathsf{Gpd}$ (in the case of schemes, replace $\mathsf{Gpd}$ by $\mathsf{Set}$), and the category of quasi-coherent modules on $\mathscr{X}$ is really just $[\mathscr{X},\mathsf{Mod}]$, the category of pseudonatural transformations from $\mathscr{X}$ to the stack of (all) modules. This definition also makes sense if we allow that $\mathscr{X} : \mathsf{CAlg}(k) \to \mathsf{CAT}$ has values in (possibly large) categories.
Usually a quasi-coherent module $M : \mathscr{X} \to \mathsf{Mod}$ is called of finite presentation if it factors through $\mathsf{Mod}_{fp}$, the stack of finitely presented modules. In my opinion, this property should be called locally of finite presentation (and some authors do this). I wonder if there is a more global condition which really means "finite presentation". For instance, consider the case that $\mathscr{X}$ is a coproduct of an infinite family of stacks $\mathscr{X}_i$. Then a quasi-coherent module $M$ on $\mathscr{X}$ should only be called (globally) of finite presentation if all almost all restrictions $M|_{\mathscr{X}_i}$ vanish, and all of them are of finite presentation.
A possible candidate is the notion of a finitely presentable object inside the category of all quasi-coherent modules. But usually these objects are not so easy to describe and their closure properties are unclear. For example, it is not clear to me if they are locally of finite presentation! I am looking either for a more concrete description of these finitely presentable objects, or for a more ad hoc finiteness condition.
My motivation: I am interested in the case $\mathscr{X}=\mathsf{Mod}_{fp}$. I have proven a classification of bounded transformations $\mathsf{Mod}_{fp} \to \mathsf{Mod}_{fp}$ (actually the unbounded case also works, but it is more ugly to state), and I would like to know if "bounded" is a special case of a more general notion of quasi-coherent modules.
