Let $\mathbf V \colon [0,T] \times \mathbb R^d \to \mathbb R^d$ (for $T>0$) be a given, bounded smooth vector field and let $\mathbf X=\mathbf X(t,x)$ be its *flow*, i.e. the unique solution to the initial-value problem
\begin{equation}
\begin{cases}
\frac{\partial}{\partial t} \mathbf X(t,x) = \mathbf V(t,\mathbf X(t,x)) & \text{ in } (0,T) \times \mathbb R^d \\
\mathbf X(0,x) = x \quad \text{ for all } x \in \mathbb R^d.
\end{cases}
\end{equation}

A well-known result in standard ODE's theory says that $$\tag{1} \nabla_x \mathbf X(t,x) = \exp\bigg( \int_0^t \nabla \mathbf V(s,\mathbf X(s,x))\,ds\bigg). $$

Is there an analogous formula to (1) for ODEs driven by (smooth) vector fields

on Riemannian manifolds? In particular, does this formula involve somehow the geometry of the Riemannian manifold? A rather precise question could be: consider the $C^1$ norm of $\mathbf X$ (or even its Lipschitz constant) w.r.t. space variable $x$: does it depend on some known tensors on the manifold (e.g. curvature)?

I have gone through books in differential geometry/differential topology (e.g. Lee, Lang) and they prove that $\mathbf X$ is smooth but do not compute explicitly the derivative.

References are very much welcome. Thanks.