**Edited**: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.

Let $N\subset M$ be a pair of the smooth manifolds and $\pi:M\to B$ a proper submersion to a smooth base. Assume moreover that the restriction $\pi_{|N}:N\to B$ is also proper and submersive, then both $\pi$ and $\pi_{|N}$ are locally trivial by applying Ehresmann's theorem separately, but I'm wondering if they can be *simultaneously* locally trivialized.

More precisely, I'm looking for a proof of the following argument:

Claim: For each $b\in B$, there is a neighborhood $U$ of $b$, and a fiber preserving diffeomorphism $$\psi:\pi^{-1}(U)\xrightarrow{\cong} M_b\times U,$$ satisfying $$\psi(N\cap \pi^{-1}(U))=N_b\times U$$ where $M_b=\pi^{-1}(b)$ and $N_b=\pi_{|N}^{-1}(b)$.

The example that I have in mind is a smooth family of pairs (cubic surface, cubic curve). Explicitly take a general pencil $\mathbb P^1$ of hyperplane sections of a smooth cubic threefold $X$ over $\mathbb C$ with base locus a cubic curve $E$ (as the intersection of any two hyperplane section in the pencil). Project the incidence variety $I=\{(y,t)|y\in X\cap H_t\}\subset X\times \mathbb P^1$ to the second factor produces a family of cubic surfaces. Now $M\to B$ will be restriction of $I\to \mathbb P^1$ to the locus where cubic surfaces are smooth and $N:=E\times B$.

(For a smooth family of compact complex manifolds, one can even ask for transversely holomorphic trivialization, see page 2 in this notes, I would also like to know if it works for a pair.)