Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known about the following 

**Question.** When can $N$ be moved by an isotopy to be nowhere-tangent to $X$? 

A necessary condition for existence of such an isotopy is that $X$ be homotopic through nowhere-zero vector fields to one nowhere-tangent to $N$. Are there invariants that can be used to imply that the desired isotopy does not exist even when this necessary condition is satisfied?