The usual Grothendieck construction is for pseudofunctors $I^{op}\to Cat$, where $I$ is a 1-category and $Cat$ is the 2-category of 1-categories. The Grothendieck construction produces a category with a functor into $I$, and the essential image is the Grothendieck fibrations over $I$.

My question: if we allow $I$ to be a bicategory, consider still $I^{op}\to Cat$. Is there a Grothendieck construction that produces a bicategory (?) with a functor into $I$? And what is the essential image of this construction (if there is one)? A quick candidate is the lax colimit of this functor, but I do not known if it is useful.

Added. We could go one step lower. Consider functor $I^{op}\to Set$, where $I$ is a 1-category. This is the same as an action of $I$. If I am not wrong, then the Grothendieck construction for $ I^{op}\to Set\to Cat$ will produces the action (translation) category.