Let $E$, $B$ be smooth manifolds, $\pi\colon E\to B$ be a smooth fiber bundle, and $h:B\to E$ a smooth embedding. I would like to learn what is known about the following

Question. When does there exist an isotopy from $h$ to a section of $\pi$?

A necessary condition for the existence of such an isotopy is that $h$ be homotopic to a section of $\pi$. Are there invariants that an isotopy does not exist even when a homotopy exists?

I am mainly interested in the special case that $B$ is a closed manifold, $E = B\times \mathbb{R}$, and $\pi:(b,t)\mapsto b$ is the trivial bundle. I would be grateful to learn about this special case or the general case above.

Let $E$, $B$ be smooth manifolds, $\pi\colon E\to B$ be a smooth fiber bundle, and $h:B\to E$ be a smooth embedding. I would like to learn what is known about the following

Question. When does there exist an isotopy from $h$ to a section of $\pi$?

A necessary condition for the existence of such an isotopy is that $h$ be homotopic to a section of $\pi$. Are there invariants that can be used to show that an isotopy does not exist even when a homotopy exists?

I am mainly interested in the special case that $B$ is a closed manifold, $E = B\times \mathbb{R}$, and $\pi:(b,t)\mapsto b$ is the trivial bundle. I would be grateful to learn about this special case or the general case above.