Conditional continuity equations pass to the marginal path

Let $\lambda$, $\pi$, and $\rho$ be as in the conditional probability-path definition, and let $u$ be a jointly measurable vector field. Suppose that for $\pi$-almost every $x_1$, $\rho$ and each component of $\rho u$ are continuously differentiable in $(t,x)$ and satisfy the classical coordinate equation ${\mathrm{\partial }}_t{\rho }_t+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}({\rho }_t{u}_t^j)=0$ on $(0,1)\times {\mathbb{R}}^d$. This is an explicit algebraic hypothesis; for general $\lambda$ it is not the definition of measure transport. Suppose also that for every compact $K\subset (0,1)\times {\mathbb{R}}^d$ there is a nonnegative $G_K\in L^1(\pi )$ which, for $\pi$-almost every $x_1$, bounds on $K$ the absolute values of ${\rho }_t(x\mid x_1)$, ${\mathrm{\partial }}_t{\rho }_t(x\mid x_1)$, each component of ${\rho }_t(x\mid x_1)u_t(x\mid x_1)$, and each derivative ${\mathrm{\partial }}_{x_j}({\rho }_t{u}_t^j)(x\mid x_1)$. If $p_t(x)>0$ on $(0,1)\times {\mathbb{R}}^d$, define $J_t(x):={\int }_{{\mathbb{R}}^d}{\rho }_t(x\mid x_1)u_t(x\mid x_1)\pi (d{x}_1),u_t(x):=\frac{J_t(x)}{p_t(x)}\mathrm{.}$ Then ${\mathrm{\partial }}_t{p}_t$ and each indicated flux derivative ${\mathrm{\partial }}_{x_j}{J}_t^j$ exist and are continuous locally in $(t,x)$, and the marginal continuity equation ${\mathrm{\partial }}_t{p}_t(x)+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}\text{\hspace{0.17em}⁣}(p_t(x)u_t^j(x))=0$ holds on $(0,1)\times {\mathbb{R}}^d$. The domination hypothesis is a sufficient Leibniz condition for the time and spatial derivatives to pass through the $x_1$ integral.

Lipman et al., §3.1 and Appendix A

Holderrieth and Erives, §3.2

The source proofs explicitly require differentiation under the endpoint integral. This item states local integrable domination for every derivative it exchanges and derives those exchanges from the mean value theorem and dominated convergence.