Wasserstein distance of conditional measures

Question

Suppose I have two measures $\mu,\nu$ defined on a common measurable space. Let $A$ be an event in the common sigma-field. Let $\mu^A(C) = \frac{\mu(A\cap C)}{\mu(A)}$ and $\nu^A(C) = \frac{\nu(A\cap C)}{\nu(A)}$ be the corresponding conditional measures. Is there a way of relating the Wasserstein distance $W_p(\mu^A,\nu^A)$ to $W_p(\mu,\nu)$? Can we say something on the optimal coupling of $\mu^A,\nu^A$ if we know the optimal coupling of $\mu,\nu$? $W_p(\mu^A,\nu^A)$ seems to be larger, since, denoting with $\pi^*$ the optimal coupling between $\mu^A,\nu^A$: $$ \int_\cal{X}\int_\cal{\tilde{X}} d(x,\tilde{x})d\pi^*(x,\tilde{x})= \frac{1}{\mu(A)}\int_A\int_\cal{\tilde{X}} d(x,\tilde{x})d\mu(x)\pi^*(\tilde{x}|x), $$ but I have no idea what I could say about the disintegration $\pi^*(\tilde{x}|x)$ once we take the "marginal" $d\mu^A$ out, could it compensate for the extra $\frac{1}{\mu(A)}$?