Let $n\in\mathbb N$, let $\mathcal A\subset \mathcal P_2(\mathbb R^n)$ be nonempty, and let $\mathcal F:\mathcal A\to(-\infty,\infty]$ be a functional. Fix $\tau>0$ and $\rho_0\in\mathcal A$ with $\mathcal F[\rho_0]<\infty$. Suppose that $(\rho_k^\tau)_{k=0}^{\infty}\subset\mathcal A$ satisfies $\rho_0^\tau=\rho_0$, $\mathcal F[\rho_k^\tau]<\infty$ for every $k\ge 0$, and, for every $k\ge 0$,

\begin{align*} \rho_{k+1}^\tau\in \operatorname*{argmin}_{\rho\in\mathcal A}\left\{\mathcal F[\rho]+\frac{1}{2\tau}W_2^2(\rho,\rho_k^\tau)\right\}. \end{align*}

Then, for every $N\in\mathbb N$,

\begin{align*} \mathcal F[\rho_N^\tau]+\frac{1}{2\tau}\sum_{k=0}^{N-1}W_2^2(\rho_{k+1}^\tau,\rho_k^\tau)\le \mathcal F[\rho_0]. \end{align*}