I came across the following question in relation to another question.
Let $X\colon M \to TM$ be a vector field over a manifold $M$ and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" for the vector field $X$, i.e. the map $\phi_t(p) = \exp_p(t X(p))$ for $p \in M$.
The maps $\phi_t$ provide two different ways to associate vectors in $T_pM$ and $T_{\phi_t(p)}M$: There is the parallel transport along the geodesic $\gamma(0) = p$, $\dot{\gamma}(0) = X(p)$ at fixed $p$, $$T_pM \ni X(p) \to \dot{\phi}_t(p) \in T_{\phi_t(p)}M.$$ And there is the pushforward $d\phi_t$ of the map $\phi_t$ at fixed $t$, $$T_pM \ni X(p) \to d\phi_t(X(p)) \in T_{\phi_t(p)}M.$$ Of course, both are applicable to arbitrary $Y \in T_pM$ as well.
Now the former is local and does not require $X$ to be a field, while the latter is nonlocal and depends on $X$ as a field. The question is whether anything can be said about the two. For example, naively I would think there is a family of vector fields $X_t$ such that $$\dot{\phi}_t = d\phi_t(X_t),$$ with $X_0 = X$. These seem interesting, since they encode the evolution of the pullback $\phi_t^* f$ of a function $f$ under $\phi_t$ (which is how they came up in my work), $$\frac{d}{dt} \phi_t^* f = X_t(\phi_t^* f).$$ There should also be fields with $X_t = X$, e.g. in flat space, so that these should contain information about the curvature.
