Consider two atomic probability measures $G_i = \sum_{j=1}^M w_{ij} \delta_{\tau_{ij}}$, $i=1, 2$, over $\mathbb R^d$. $M$ could be either finite or infinite. The weights $w_{ij}$ are nonnegative and $\sum_{j=1}^M w_{ij} = 1$, $\tau_{ij} \in \mathbb R^d$.
Now consider convolving each measure with a smooth kernel $K(\cdot, \cdot)$ such that $K(\cdot, \tau)$ is a probability density over $\mathbb R^d$ for each value of $\tau$. For instance, $K$ could be the Gaussian kernel with fixed covariance matrix $\Sigma$ (symmetric and positive definite): $$ K(x, \tau) = K_\Sigma(x, \tau) = \frac{1}{(2\pi)^{d/2} |\Sigma|} \exp((x-\tau)^\prime \Sigma^{-1} (x-\tau)) $$
Define the probability densities over $\mathbb R^q$ $$ p_i(\cdot) = \int_{\mathbb R^q} K(\cdot, \tau) G_i(d\tau) = \sum_{j=1}^M w_{ij} K(\cdot, \tau_{ij}) $$
I'm interested in estimating a bound of the kind $$ W_2^2(p_1, p_2) \ge c W_2^2(G_1, G_2) $$ for $c$ a constant depending on the parameters in $K(\cdot, \cdot)$ and $W_2^2$ the squared 2-Wasserstein distance.
Is there some reference to problems of this kind?