Let $\mathbf{Top}$ denote the $2$-category of topological spaces, continuous mappings, and homotopies between them. Let $\mathbf{C}$ denote a wide subcategory of $\mathbf{Top}$. Then we get a wide sub-$2$-category $\overline{\mathbf{C}}$ of $\mathbf{Top}$ as follows. Firstly, the underlying category of $\overline{\mathbf{C}}$ is just $\mathbf{C}$. Secondly, the $2$-cells are precisely those homotopies between $\mathbf{C}$-arrows that "remain" in $\mathbf{C}$ through the entire deformation process. For example, if $\mathbf{C}$ is the wide subcategory of $\mathbf{Top}$ where the morphisms are injective continuous functions, then a $\mathbf{C}$-morphism $X \rightarrow Y$ is an "$X$-shaped" knot in the space $Y$, and the $2$-cells of $\overline{\mathbf{C}}$ are "witnesses" that two knots are equivalent. Take $X$ to be the circle and $Y$ to be $\mathbb{R}^3$ to recover the usual definitions.
Question. What, if anything, does higher category theory have to say about $2$-categories $\mathbf{T}$ equipped with a special way of extending each subcategory $\mathbf{C}$ into a sub-$2$-category $\overline{\mathbf{C}}$?
E.g. What are the appropriate axioms, definitions, terms and phrases, generalizations, etc.