Let $n\in\mathbb N$, let $\lambda\in\mathbb R$, and let $\mathcal E:\mathcal P_2(\mathbb R^n)\to(-\infty,+\infty]$ be a proper functional, with nonempty effective domain $\operatorname{Dom}(\mathcal E):=\{\mu\in\mathcal P_2(\mathbb R^n):\mathcal E[\mu]<\infty\}$. Let

\begin{align*} \rho:[0,\infty)\to\mathcal P_2(\mathbb R^n) \end{align*}

and

\begin{align*} \eta:[0,\infty)\to\mathcal P_2(\mathbb R^n) \end{align*}

be locally absolutely continuous curves, written $t\mapsto\rho_t$ and $t\mapsto\eta_t$, such that $\rho_t,\eta_t\in\operatorname{Dom}(\mathcal E)$ for every $t>0$. Assume that both curves are strong $\operatorname{EVI}_\lambda$ gradient flows of the same functional $\mathcal E$ in the following sense: for every compact interval $I\subset(0,\infty)$ there are Borel sets $A_\rho(I),A_\eta(I)\subset I$ with $\mathcal L^1(I\setminus A_\rho(I))=\mathcal L^1(I\setminus A_\eta(I))=0$ such that, for every $t\in A_\rho(I)$ and every $\sigma\in\operatorname{Dom}(\mathcal E)$, the derivative of $s\mapsto W_2^2(\rho_s,\sigma)$ exists at $s=t$ and

\begin{align*} \frac{1}{2}\frac{d}{dt}W_2^2(\rho_t,\sigma)+\frac{\lambda}{2}W_2^2(\rho_t,\sigma)\le \mathcal E[\sigma]-\mathcal E[\rho_t]. \end{align*}

Likewise, for every $t\in A_\eta(I)$ and every $\sigma\in\operatorname{Dom}(\mathcal E)$, the derivative of $s\mapsto W_2^2(\eta_s,\sigma)$ exists at $s=t$ and

\begin{align*} \frac{1}{2}\frac{d}{dt}W_2^2(\eta_t,\sigma)+\frac{\lambda}{2}W_2^2(\eta_t,\sigma)\le \mathcal E[\sigma]-\mathcal E[\eta_t]. \end{align*}

Then, for every $t\ge 0$,

\begin{align*} W_2(\rho_t,\eta_t)\le e^{-\lambda t}W_2(\rho_0,\eta_0). \end{align*}