Suppose $V = V^i \dfrac{\partial}{\partial x^i}$ is a vector field in $\mathbb R^n$. Let $\theta_t : \mathbb R^n \to \mathbb R^n$ denote the local flow generated by $V$. Suppose $p \in \mathbb R^n$, and let $w = w^i_0 \dfrac{\partial}{\partial x_i} \in T^p$. Suppose $w(t) = w^i(t) \dfrac{\partial}{\partial x^i}\bigg|_{\theta_t(p)}$ is a vector field along the integral curve $\theta_t(p)$ whose coordinates satisfy the system of differential equations \begin{equation} \begin{cases} \dot w^i(t) = w^j(t) \dfrac{\partial V^i}{\partial x^j}\left(\theta_t(p)\right), \\ w^i(0) = w^i_0 \end{cases} \end{equation} Question 1: Is the solution to this initial value problem given by $w(t) = d\left(\theta_t\right)_p w$?
It seems a little too good to be true, since the coordinates of $w(t) = d(\theta_t)_p w$ are $$ w^i(t) = w^j\frac{\partial \theta_t^i}{\partial x^j}\left(\theta_t(p)\right), $$ where $w^j = w^j(0) = $ const, and so the time derivatives of these coordinates would be $$ \dot w^i(t) = w^j \frac{\partial V^i}{\partial x^j}\left(\theta_t(p)\right) $$ which is close to, but quite different from, the thing I'd like to show. So is there some other variational relationship between the action of $d(\theta_t)_p$ and the partial derivatives of the vector field?
Related to this, it seems this system of differential equations can be expressed on an arbitrary Riemannian manifold as $$ D_t w = \nabla_w V, $$ where $D_t w$ is the covariant derivative of the vector field $w(t)$ along the curve $\theta_t(p)$, whose infinitesimal generator is $V$. Since the Christoffel symbols of the Levi-Civita connection $\nabla$ are symmetric in the lower indices (i.e. $\Gamma_{ij}^k = \Gamma_{ji}^k$ in any coordinates), the coordinate expression of $D_t w = \nabla_w V$ are equivalent to the system of ODEs described above.
Question 2: If $V$ is a flow on a Riemannian manifold $(M,g)$, and $w(t)$ is a vector field along an integral curve $\theta_t(p)$ for $V$ which satisfies the initial value problem $D_t w = \nabla_w V$, $w(0) = w \in T_p M$, is it the case that $w(t) = d(\theta_t)_p w$? If not, is there a variational relationship between $\nabla V$ and the action of $d(\theta_t)$?