  Let $(M^n,g,f)$ be a Ricci shrinker such that $\textrm{Ric}_f=\frac{1}{2}g$
and the measure induced by the weighted volume element
$(4\pi)^{-\frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point
$p\in M$, we consider two probability measures defined in the tangent space
$T_pM$, namely the Gaussian measure $\gamma$ and the measure $\overline{\nu}$
induced by the exponential map of $M$ to $p$. In this paper, we prove a result
that provides an upper estimate for the Wasserstein distance with respect to
the Euclidean metric $g_0$ between the measures $\overline{\nu}$ and $\gamma$,
and which also elucidates the rigidity implications resulting from this
estimate.
