Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface, 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 $X$?

In particular, are there invariants that can be used to imply a negative answer even when $X$ is homotopic through nowhere-zero vector fields to one nowhere-tangent to $N$?

Edit: I am mainly interested in the case that $M$ is noncompact without boundary, in particular the case that there is a diffeomorphism from $M$ to $N\times \mathbb{R}$ sending $X$ to the vector field $(0,1)$ tangent to curves of the form $\{m\}\times \mathbb{R}$. Also, I am mainly interested in the case that every component of $M\setminus N$ is noncompact.