be a smooth family of positive probability densities on $\mathbb R^n$ such that $\rho_t\,d\mathcal L^n\in\mathcal P_2(\mathbb R^n)$ for every $t\in I$. Let
Assume that $\rho_t$ and $\phi_t$ have sufficient decay at infinity to justify the integrations by parts defining the distributional continuity equation, and that
Assume finally that, for $\mathcal L^1$-a.e. $t\in I$, $\nabla\phi_t$ is the $L^2(\mu_t;\mathbb R^n)$-minimal velocity representative among all Borel vector fields generating the same density derivative $\partial_t\rho_t$ through the continuity equation. Then the curve
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\begin{align*}
t\mapsto \mu_t
\end{align*}
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is absolutely continuous in $(\mathcal P_2(\mathbb R^n),W_2)$ locally on $I$, and for $\mathcal L^1$-a.e. $t\in I$ its Wasserstein metric speed satisfies