Let $I\subset\mathbb R$ be an open interval, let $\mathcal A\subset C^\infty(\mathbb R^n)$ be an admissible class of strictly positive probability densities, and let $\mathcal F:\mathcal A\to\mathbb R$ be a functional with first variation on $\mathcal A$. More precisely, assume that for every $\rho\in\mathcal A$ there is a smooth function $\frac{\delta\mathcal F}{\delta\rho}(\rho):\mathbb R^n\to\mathbb R$ such that for every open interval $J\subset\mathbb R$, every smooth curve $\eta:J\times\mathbb R^n\to(0,\infty)$ with $\eta_s:=\eta(s,\cdot)\in\mathcal A$ for all $s\in J$, and every $s_0\in J$, one has, with $\sigma_{s_0}:\mathbb R^n\to\mathbb R$ defined by $\sigma_{s_0}(x)=\partial_s\eta(s_0,x)$,

\begin{align*} \left.\frac{d}{ds}\right|_{s=s_0}\mathcal F[\eta_s]=\int_{\mathbb R^n}\frac{\delta\mathcal F}{\delta\rho}(\eta_{s_0})(x)\,\sigma_{s_0}(x)\,d\mathcal L^n(x). \end{align*}

Let $\rho:I\times\mathbb R^n\to(0,\infty)$ and $\phi:I\times\mathbb R^n\to\mathbb R$ be smooth maps such that $\rho_t:=\rho(t,\cdot)\in\mathcal A$ for every $t\in I$. Assume that, for every $t\in I$, the first-variation potential $\psi_t:\mathbb R^n\to\mathbb R$ defined by $\psi_t(x)=\frac{\delta\mathcal F}{\delta\rho}(\rho_t)(x)$ and the flux $J_t:\mathbb R^n\to\mathbb R^n$ defined by $J_t(x)=\rho_t(x)\nabla\phi_t(x)$ satisfy the integration-by-parts identity

\begin{align*} -\int_{\mathbb R^n}\psi_t(x)\,\operatorname{div}J_t(x)\,d\mathcal L^n(x)=\int_{\mathbb R^n}\nabla\psi_t(x)\cdot J_t(x)\,d\mathcal L^n(x), \end{align*}

for instance because the relevant functions have compact support in the spatial variable or sufficient decay at infinity. Suppose that $(\rho_t,\phi_t)$ satisfies the continuity equation

\begin{align*} \partial_t\rho_t+\operatorname{div}(\rho_t\nabla\phi_t)=0 \end{align*}

on $I\times\mathbb R^n$. Then, for every $t\in I$,

\begin{align*} \frac{d}{dt}\mathcal F[\rho_t]=\int_{\mathbb R^n}\nabla\left(\frac{\delta\mathcal F}{\delta\rho}(\rho_t)\right)(x)\cdot\nabla\phi_t(x)\,\rho_t(x)\,d\mathcal L^n(x). \end{align*}