Let $M$ be a (smooth) compact manifold and $X$ be a vector field on $M$. Let $\psi:\mathbb{R}\times M\to M$ be the flow associated to $X$. We know that:
for each $t\in\mathbb{R}$, $\psi|_{\{t\}\times\,M}$ is a diffeomorphism,
for each $p\in M$ such that $X_p\neq 0$, $\psi|_{\mathbb{R}\times\{p\}}$ is an immersion.
Are there other kown results which gives you information about the flow for other submanifolds of $\mathbb{R}\times M$ (such as $\mathbb{R}\times S$ where $S$ is a submanifold of $M$)? Thank you for your answers!
One such result is the flowout theorem, which is Theorem 9.20 in my Introduction to Smooth Manifolds (2nd ed.). It says, in part, that if $S$ is a $k$-dimensional embedded submanifold of $M$ and $X$ is nowhere tangent to $S$, then the restriction of $\psi$ to $\mathbb R\times S$ is an immersion, and its image is an immersed $(k+1)$-dimensional submanifold near $S$.