Let $n\in\mathbb N$, let $\lambda\in\mathbb R$, and let $\mathcal P_2(\mathbb R^n)$ denote the set of Borel probability measures on $\mathbb R^n$ with finite second moment, equipped with the quadratic Wasserstein distance $W_2$. For $\mu,\nu\in\mathcal P_2(\mathbb R^n)$, let $\Gamma_o(\mu,\nu)$ denote the set of optimal transport plans between $\mu$ and $\nu$ for the quadratic cost $|x-y|^2$. Let $\mathcal E:\mathcal P_2(\mathbb R^n)\to(-\infty,\infty]$ be a proper lower semicontinuous functional that is $\lambda$-convex along constant-speed $W_2$-geodesics. Define $D(\mathcal E):=\{\mu\in\mathcal P_2(\mathbb R^n):\mathcal E[\mu]<\infty\}$. Let $\rho:(0,\infty)\to\mathcal P_2(\mathbb R^n)$, $t\mapsto\rho_t$, be locally absolutely continuous as a curve in $(\mathcal P_2(\mathbb R^n),W_2)$. Assume that $\rho_t\in D(\mathcal E)$ for $\mathcal L^1$-a.e. $t>0$ and that there exists a Borel vector field $v:(0,\infty)\times\mathbb R^n\to\mathbb R^n$, written $v_t(x):=v(t,x)$, such that $v_t\in L^2(\mathbb R^n,\mathcal B(\mathbb R^n),\rho_t;\mathbb R^n)$ for $\mathcal L^1$-a.e. $t>0$, and such that for every compact interval $[a,b]\subset(0,\infty)$ one has \begin{align*}\int_a^b\int_{\mathbb R^n}|v_t(x)|^2\,d\rho_t(x)\,d\mathcal L^1(t)<\infty.\end{align*} Assume that $\partial_t\rho_t+\nabla\cdot(v_t\rho_t)=0$ in the sense of distributions on $(0,\infty)\times\mathbb R^n$. Assume moreover that $v_t$ is the tangent velocity of the Wasserstein absolutely continuous curve $\rho$ for $\mathcal L^1$-a.e. $t>0$, meaning that $v_t$ belongs to the $L^2(\rho_t;\mathbb R^n)$-closure of gradients of smooth compactly supported functions and $\|v_t\|_{L^2(\rho_t)}=|\rho'|(t)$ for $\mathcal L^1$-a.e. $t>0$, where $|\rho'|(t)$ is the metric derivative of $\rho$. Assume also that $-v_t$ belongs to the strong Wasserstein subdifferential $\partial\mathcal E(\rho_t)$ in the following sense: for $\mathcal L^1$-a.e. $t>0$, for every $\nu\in D(\mathcal E)$, and for every optimal transport plan $\gamma\in\Gamma_o(\rho_t,\nu)$, \begin{align*}\mathcal E[\nu]-\mathcal E[\rho_t]\ge \int_{\mathbb R^n\times\mathbb R^n}(-v_t(x))\cdot(y-x)\,d\gamma(x,y)+\frac{\lambda}{2}W_2(\rho_t,\nu)^2.\end{align*} Then, for every $\nu\in D(\mathcal E)$, the function $f_\nu:(0,\infty)\to[0,\infty)$, $t\mapsto W_2(\rho_t,\nu)^2$, is absolutely continuous on every compact subinterval of $(0,\infty)$, and for $\mathcal L^1$-a.e. $t>0$, \begin{align*}\frac{1}{2}\frac{d}{dt}W_2(\rho_t,\nu)^2+\frac{\lambda}{2}W_2(\rho_t,\nu)^2+\mathcal E[\rho_t]\le \mathcal E[\nu].\end{align*}