Given two schemes $X$ and $Y$ one can consider additive functors (eventually with some nice additional property) between the categories of $\mathcal{O}_X$-modules and of $\mathcal{O}_Y$ modules. Among these one has those "of a geometric nature", in particular Furier-Mukai-type functors, i.e., those of the form $\Phi_{Z,\mathcal{P}}:=g_*(\mathcal{P}\otimes f^*-)$, where $(f,g):Z\to X\times Y$ and $\mathcal{P}$ is some "twisting" sheaf on $Z$. With suitable finiteness assumptions (e.g., if $Z$ is Noetherian), $\Phi_{Z,\mathcal{P}}$ is a functor $QC(X)\to QC(Y)$. Also, one can consider a derived variant of this, and look at $\Phi_{Z,\mathcal{P}}$ as to a functor $D^b(Coh(X))\to D^b(Coh(Y))$ (all this is very well known and studied, I'm writing it only to provide a setting for my question).

Given two Furier-Mukai-type functors $\Phi_{Z,\mathcal{P}}$ and $\Phi_{W,\mathcal{Q}}$ between categories of $\mathcal{O}_X$-modules and of $\mathcal{O}_Y$-modules, are there natural "geometric" transformations between these functors? which are the main examples of these? (here "geoemtric" is somehow vague: it pretends to mean "something with the same flavor of Fourier-Mukai transform, e.g., maybe something involving a morphism $T\to Z\times W$ and some twisting kernel over $T$)