My question is:

Has anyone constructed an $(\infty,2)$-category whose objects are (projective, maybe smooth, ...) varieties, and where the 1-morphisms from $X$ to $Y$ are given by $D^b_\infty\text{Coh}(X \times Y)$ (an $(\infty,1)$-enhancement of $D^b\text{Coh}(X \times Y)$)?

By "$(\infty,1)$-enhancement" I mean some $(\infty,1)$-category whose homotopy category is $D^b\text{Coh}$.

I would hope that binary composition would descend to the functor $$D^b\text{Coh}(Y\times Z) \times D^b\text{Coh}(X \times Y) \to D^b\text{Coh}(X \times Z)$$ that sends $(Q,P)$ to $\pi_{02,*}(\pi_{01}^*P \otimes \pi_{12}^*Q)$ a la the Fourier--Mukai transform (where tensor and the projections are the derived functors).

If the answer is "yes", I wonder whether this is special to derived categories of coherent sheaves or whether it's a more general algebraic result about collections of $(\infty,1)$-categories of a certain kind?

I'm new to infinity infinity stuff and I might have misused a technical term, so it might be best not to interpret my words too literally.