For a (discrete) measure $G$ on some reasonable metric space $\Theta$, consider the map $G \mapsto f_G$ defined as $$ f_G := f*G(dx) := \int f(dx|\theta) G(d\theta) $$ for some nice kernel function $f(dx|\theta)$ on $X\times \Theta$ where $X$ is some other reasonable metric space. We think of $f_G$ as a measure on $X$, which is a smoothed version of $G$. For simplicity one can think of $X = \mathbb{R}^d$ and $f(dx|\theta) = f(x|\theta) dx$ for some density $f(x|\theta)$ with respect to Lebesgue measure. We can even restrict to the case $f(x|\theta) = f(x-\theta)$ assuming $X =\Theta = \mathbb{R}^d$.

Let us equip both the spaces of measures on $\Theta$ and $X$ with some Wasserstein distance, say $W_2$. The question is how to obtain a bound of the form $$ \alpha(W_2(G,G')) \le W_2(f_G,f_{G'}), \quad \forall G,G' $$ for some $\alpha : [0,\infty) \to [0,\infty)$. In other words, we are interested in the modulus of continuity of the "inverse" of $G \mapsto f_{G}$. One can even consider the simplest case where $G = \delta_{\theta}$ and $G' = \delta_{\theta'}$ so that $W_2(G,G') = d_{\Theta}(\theta,\theta')$.