Let $C$ be a small category and $M$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $Fun(C,M)$ of functors from $C$ to $M$, all with the same (levelwise) weak equivalences. The whole point of having such a model structure is that it should present the $\infty$-category $Fun(C,\tilde M)$, where $\tilde M$ is the $\infty$-category presented by $M$. But I'm not sure when this is actually the case.

Of course, by "the $\infty$-category presented by $M$", I mean $\tilde M = M[W^{-1}]$ is $M$ localized at the weak equivalences in the $\infty$-categorical sense, and similarly "the $\infty$-category presented by $Fun(C,M)$" is the $\infty$-categorical localization $Fun(C,M)[Fun(C,W)^{-1}]$.

**Questions:**

If $C$ is a small category and $M$ is a model category, then under what conditions do the standard model structures on $Fun(C,M)$ present the $\infty$-category $Fun(C,\tilde M)$, where $\tilde M = M[W^{-1}]$ is the $\infty$-category presented by $M$?

More generally, if $C$ and $M$ are relative categories, then under what conditions does the mapping relative category $\widetilde{Fun(C,M)} = Fun(\tilde C, \tilde M)$ where $\tilde{(-)}$ denotes taking the associated quasicategory?

In a more model-independent direction, when does localization of $\infty$-categories commute with taking functor categories? That is, when does $Fun(C,M[W^{-1}]) = Fun(C,M)[Fun(C,W)^{-1}]$ where $C,M$ are $\infty$-categories and $W \subseteq M$ is a subcategory?