Given a set valued function $F$ such that for every $x\in M$ (a manifold) we have that $F(x)\subset T_xM$, a differential inclusion is the "equation", $\dot{x} \in F(x)$.

I was wondering if someone has studied an analog of this for distributions, that is, consider a function such that $F(x)\subset (T_xM)^k$ and search for foliations such that in every point are tangent to the span of an element of $F(x)$.

I've tried googleing some similar names and could not find anything, but maybe here someone knows the key-word I am looking for. Also if there is any reference that would be good.