For a vector field $X$ on a manifold there are two ways to define a Lie derivative: an algebraic one using Cartan's formula $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$ and a dynamical one using the flow $\phi^X_t$, $${\cal L}_X \alpha = \frac{d}{dt}\biggl|_{t=0} \left(\bigl(\Phi^t_X\bigr)^* T\right)$$
Given a multivector field $X$ one can define its Lie derivative by means of Cartan formula, i., $\mathcal{L}_X \alpha = i_X d \alpha + d i_X alpha$$\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$. See, for example [1].
My broad question is if the Lie derivative by a multivector means something dynamical. These are some fuzzy questions for which any help or reference would be welcomed.
In the simplest case, if $X = X_1\wedge \ldots \wedge X_n$ and the e_i span an integrable distribution, is $\alpha$ constant in some sense on the leaves of this distribution. What happens if the multivector is not integrable?
Is there a generalization of the concept of flow for a multivector that applies to this situation?
My current geometric understanding of general multivectors is that they are linear combinations of hyperplanes modulo the Plücker relations (whose geometric interpretation feels somewhat obscure to me). I would like to have a better interpretation. On question [2] on this site it is stated that they are global sections of a line bundle over the Grasmannian, some reference of this fact would be useful.
[2] Denis Serre (https://mathoverflow.net/users/8799/denis-serre), Grassmannian as a submanifold of $\Lambda^m(E)$., URL (version: 2011-11-14): Grassmannian as a submanifold of $\Lambda^m(E)$.
