Is the following fact known? If yes - what is the reference?
Let $\phi:X\to Y$ be a submersion of smooth manifolds with connected fibers. Let $s_0,s_1:Y\to X$ be its (smooth) sections. Then, for any $y\in Y$, there exists an open neighborhood $U$ of $y$ and a smooth homotopy $h:[0,1] \times U\to X$ such that $h|_{\{0\}\times U}=s_0$, $h|_{\{1\}\times U}=s_1$, and $h|_{\{t\}\times U}$ is a section of $\phi$ for any $t$.
We seem to have a proof, but we prefer to give a reference.