Let $\Omega$ be a open, connected domain such that $\overline\Omega$ is compact in a Riemannian manifold $(M,g)$. Let $V$ be a smooth vector field on $M$, define $$ \Omega_{\varepsilon}=\{ x\in\Omega\ :\ \text{dist}(x,\partial\Omega)>\varepsilon \}. $$
Let $f\in L^p(\Omega;\Bbb R)$, where $p\in[1,\infty)$. For a sufficiently small $\delta>0$, the map $x\mapsto \exp(x,\delta V)$ is a diffemorphism from $\Omega_{\varepsilon}$ onto its image that is also contained in $\Omega$.
Is it possible to find a constant $C>0$ such that $$ \int_{\Omega_{\varepsilon}}|f(x)-f(\exp(x,\delta V))|^p dx < C $$ uniformly as $\delta\to0$?
Any help/suggestion is very appreciated.
PS: This seems similar to the continuity of translation in $L^p$ for Euclidean domain. Although knowing that it is uniformly bounded is enough for me, it'd be nice to know if it is possible to replace $C$ with $O(\delta)$ as well.