Let $\mathsf{Stoch}$ denote the Kleisli category of the Giry monad. That is, $\mathsf{Stoch}$ is a category whose objects are measurable spaces and for which a morphism $f\in\mathsf{Stoch}(X,Y)$ is a map sending points in $X$ to probability measures on $Y$; see nLab: Giry monad for details.

Let $\mathsf{Met}$ denote the symmetric monoidal category of metric spaces $(X,d)$ and short maps (functions $f\colon X\to Y$ satisfying $d(x_1,x_2)\geq d(fx_1,fx_2)$ for all $x_1,x_2\in X$).

Question: Is there a way to enrich $\mathsf{Stoch}$ in $\mathsf{Met}$?

In other words, I'm looking for a way to endow each hom-set $\mathsf{Stoch}(X,Y)$ with a metric space structure in such a way that the composition operation is short: $$ d(f_1,f_2)+d(g_1,g_2) \geq d(g_1\circ f_1, g_2\circ f_2) $$

The upshot would be that when working in $\mathsf{Stoch}$, diagrams would not simply either commute or not commute; they would commute up to some distance, and this distance would behave sensibly under "diagram chasing".