Kan extensions make sense in any 2-category. I am interested in Kan extensions in "concrete" 2-categories consisting of actual categories with some sort of structure (e.g., finite products, finite limits, pretoposes, toposes, etc.), with 1- and 2-cells structure-preserving functors and any natural transformations.

First of all, when do such extensions exist? The usual Kan extensions exist whenever the target category is co-complete, but these functors do not typically preserve the relevant structure (aside: what are some relatively simple examples of this phenomenon). Is there some way to "fix up" this problem and provide a "structure-preserving Kan" given co-completeness? If not, are there other conditions which might allow us to do that? Failing either of these, are there at least conditions under which the ordinary Kan extensions will preserve structure X?

This leads to a second question. If (some) extensions do exist in a concrete 2-category, how are they related to the usual extensions in $\textbf{Cat}$? Intuitively, I would expect the forgetful functor to preserve right Kan extensions, as these are akin to limits, and the free completion to preserve left Kan extensions.

Thanks!