Consider a smooth vector bundle $\pi: E\rightarrow M$, the associated infinite jet bundle $J^\infty(\pi)$, and evolutionary vector fields $\partial_\varphi = \sum_{i,\sigma}(D_\sigma\varphi^i)\frac{\partial}{\partial u^i_\sigma}$. Here $D_\sigma$ is the composition of total derivatives corresponding to the multi-index $\sigma$. As is well known, these are the vector fields which leave the Cartan distribution invariant and are vertical.

The question is this: what does the (local) flow of such a vector field look like? (Recall that the flow of a vector field $V$ is a map $\theta$ from a subset of $\mathbb{R} \times M$ to $M$ such that $\left.\frac{d}{dt}\right|_{t=0}\theta(\cdot,x) = V_x$ for all $x$.) I have read at various places (e.g. *Symmetries and Conservation Laws for Differential Equations of Mathematical Physics* by Krasil'shschik and Vinogradov, and here) that in the case of evolutionary vector fields, these flows are sections of the bundle $\pi$, i.e. smooth maps $s: M \rightarrow E$ such that $\pi\circ s = \text{id}$. (Thus these evolutionary vector fields gain the interpretation as specifying the evolution of sections of the bundle.)

Can anyone explain how this works? Why are these flows in fact sections of the bundle?