I would like to know if there exists a satisfying generalization of higher-dimensional category theory on objects, that doesn't forget the inner structure of objects. Usually, what people do is to look for higher dimensional arrows, thus remembering the relations between 0-cells but forgetting that $0$-cells can be "made up".

What I want is to "keep track" of the inner structure of my objects (for example, they can be subcategories or categories).

Example: Cat as a 2-category is such that its 0-cells are small categories, 1-cells are functors, 2-cells are natural transformations. As an abstract 2-category, Cat doesn't "see" the inner structure of 0-cells and treat them as pure syntax. It is as if there was some "forgetful inner structure functor" that was applied to a higher dimensional analog on object. The same happens when one construct a category where objects are diagrams of another category. Indeed, let $\mathcal{C}$ be an abstract category. Let $Grp(\mathcal{C})$ be the category where objects are group objects of $\mathcal{C}$ and arrows are internal homomorphism. In this example, one is once again forgetting the inner structure of group object and see them as 0-cell (i.e., point on the underlying graph).

More formally, one can call $2*$-category a "collection" of categories + functors between them (in order to mimic a subcategory of Cat where one remembers the inner structure of objects). A $2*$-functor would take as input objects, arrows, and functors of a $2*$-category, and maps them respectively to objects, arrows, and functors with obvious axioms (preservation of composition of functors + unit). The "forgetful inner structure functor" is then the $2*$-functor injective on functors, sending all objects of a given $2*$-category to a constant object of the same category, all arrows to the identity, but keeping tracks of the functors.

PS: my formalization is just done here to clarify the idea, its obviously not the way to go because it doesn't encompass the example of $Grp(\mathcal{C})$. Actually, it would be an even higher dimensional analog. A group object in $\mathcal{C}$ is already a $2*$-category: it is a functor from the sketch of a group object to $\mathcal{C}$, and arrows between them is given by natural transformations between such functors. So basically, $Grp(\mathcal{C})$ is a pretty good example of mix between higher-dimensional category theory both on objects and arrows.