If we have a Riemann manifold $M$ with metric $g_{\mu\nu}$ and with reference measure $e^{-\psi(x)}\sqrt{g}dx$. We define a function $\rho(x)$ on $M$ which will be our probability density, from this we define a new probability measure $d\mu = \rho e^{-\psi(x)}\sqrt{g}dx$. The function $\rho$ can evolve (smoothly) along a curve parametrised by $t$, therefore also $\mu$ evolves, so we will write $\mu_t$ for $t\neq 0$.
Let's take the following ansatz, with $\zeta$ and $\eta$ being two smooth functions on $M$ $$ \int_M \nabla \zeta \cdot \nabla \eta d\mu = \frac{d}{dt}\Big |_{t=0} \int_M \zeta d\mu_t $$ From this we want to derive the following continuity equation for $\mu$ $$ \frac{\partial \mu}{\partial t} = -\nabla \cdot \left(\mu\nabla\eta\right) $$
I have done the following calculations $$ \frac{d}{dt}\Big |_{t=0} \int_M \zeta d\mu_t = \int_M \zeta \frac{\partial}{\partial t}\Big |_{t=0}\mu_t dx = \int_M \zeta \frac{\partial}{\partial t}\Big |_{t=0}\rho_t e^{-\psi} \sqrt g dx $$ $$ \int_M \nabla \zeta \cdot \nabla \eta d\mu = \int_M (\partial_\nu \zeta \partial^\nu \eta )\mu dx = -\int_M \zeta \partial_\nu(\mu \partial^\nu\eta ) dx = -\int_M \zeta \nabla_\nu(\rho e^{-\psi} \nabla^\nu\eta ) \sqrt g dx $$ therefore we can get a covariant (with respect to the manifold $M$) continuity equation only for $\rho e^{-\psi}$ and not $\mu$ $$ \frac{\partial}{\partial t}\rho e^{-\psi} = - \nabla_\nu(\rho e^{-\psi} \nabla^\nu\eta ) $$ even worse if our theory has non-zero torsion tensor $T_{\mu\nu}{}^\rho$, because we would get $$ \frac{\partial}{\partial t}\rho e^{-\psi} = - \nabla_\nu(\rho e^{-\psi} \nabla^\nu\eta ) + T_{\nu\mu}{}^\nu (\rho e^{-\psi}\nabla^\nu \eta) $$ If we really want to write this continuity equation for $\mu$, from the above calculations I get $$ \frac{\partial \mu}{\partial t} = -\partial_\nu\left(\mu\partial^\nu\eta\right) $$ and not $$ \frac{\partial \mu}{\partial t} = -\nabla \cdot \left(\mu\nabla\eta\right) $$ I think there is something conceptual that I am missing, if anyone is wondering where all this comes from I was reading Chap. 15 of "Optimal transport, old and new" by Cédric Villani, where the covariant continuity equation for $\mu$ is needed there.