I know about the flowout of a vector field along a smooth submanifold (as in, e.g., Lee, Introduction to Smooth Manifolds, Theorem 9.20), but I came a cross a similar notion for distributions and I'm not entirely sure what it means.
More precisely, let $M$ be a smooth manifold, $D \subseteq TM$ an involutive distribution, and $S$ a smooth embedded submanifold. What is:
The flowout of $D$ along $S$?
My guess is that, first, $S$ has to be such that $T_pS \cap D_p = 0$ for all $p \in S$ and then, by choosing local frames for $D$ and combining their flowouts, we get a $\dim(S) + \mathrm{rank}(D)$ dimensional submanifold of $M$ containing $S$. Is that more or less the idea? What is the precise formulation? Any reference?