Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup\_{x \in X} d\_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in Wasserstein distance; i.e.:
$$
W\_1\left(f\_{\#}\mathbb{P},g\_{\#}\mathbb{P}\right)<\delta(\epsilon)
,
$$
for every $\mathbb{P}\in \mathcal{P}\_1(X)$ for some $\delta(\epsilon)$ depending on $\epsilon,f,g$ but for which $\lim\limits\_{\epsilon \to 0} \delta(\epsilon)=0$?



Continuity of pushforward operation
Yes. If the uniform distance of $f$ and $g$ is less than $\epsilon$, simply take your coupling to be the push-forward of the function $x\mapsto\big(f(x),g(x)\big)$. The resulting coupling verifies that the Wasserstein-$1$-distance is less than $\epsilon$.


