For smooth projective varieties it is known that $$D^b_\infty Coh(X\times Y)\simeq Hom(D^b_\infty Coh(X), D^b_\infty Coh(Y))$$ compatibly with the composition you describe (note here $D^b_\infty Coh$ coincides with the $\infty$-category $Perf$ of perfect complexes). Hence your desired $(\infty,2)$-category is a full subcategory of the $(\infty,2)$-category of dg categories (or small idempotent-complete stable $\infty$-categories) where the above Homs take place. The analogous result for $QCoh$ holds in much greater generality (without smoothness or projectivity - eg for quasicompact quasiseparated schemes).

For smooth projective varieties it is known that $$D^b_\infty Coh(X\times Y)\simeq Hom(D^b_\infty Coh(X), D^b_\infty Coh(Y))$$ compatibly with the composition you describe (note here $D^b_\infty Coh$ coincides with the $\infty$-category $Perf$ of perfect complexes). Hence your desired $(\infty,2)$-category is a full subcategory of the $(\infty,2)$-category of dg categories (or small idempotent-complete stable $\infty$-categories) where the above Homs take place. The analogous result for $QCoh$ holds in much greater generality (without smoothness or projectivity - eg for quasicompact quasiseparated schemes).

Edit: Without smoothness, you find (here) that $D^b_\infty Coh(X\times Y)$ represents functors from $Perf(X)$ to $D^b_\infty Coh(Y)$, and that functors from $D^b_\infty Coh(X)$ to $D^b_\infty Coh(Y)$ are represented by kernels which are coherent relative to the first factor. Note you have to be careful in the formulation of the question since $D^b_\infty Coh$ is not preserved by tensor product on a singular variety! (e.g. self-tor of a skyscraper at a singular point is not bounded, i.e., not in $D^b_\infty Coh$)