Can we characterize entirely for which categories $C$ the end on $C$ preserves filtered colimits, in a sense that the natural map

$$ \operatorname*{colim}_{i \in I} \int_{x \in C} A_i (x,x) \to \int_{x \in C} \operatorname*{colim}_{i \in I} A_i (x,x)$$

is an isomorphism for every filtered diagram of $A_i \in \text{Set}^{C^\text{op} \times C}$ ?

It is easy to see that this will be the case when $I$ is a finitely generated category, but are there some non-finitely generated categories for which this is also true?

Of course, if one replace ends by limits then there are plenty of example: for any category with an initial objects $C$, limits along $C$ are just evaluation at the initial object and these preserves all colimits, and more generally it is enough for the category to admit a cofinal finitely generated subcategory but I'm not aware of a similar phenomenon for ends...