Let $A,B \subset M$ be two transversal submanifolds of a compact manifold $M$. It seems rather intuitive that if $A$ and $B$ are deformed (say smoothly) in a way that they remain transversal to each other, the homotopy type of the intersection remains unchanged during the deformation.

I would like to know if there is a way to write such a homotopy in the following situations:

1. $A = f^{-1}(0)$ is the zero level set of some smooth function $f : M \to \mathbb{R}$, and $B = B_s$ is a smooth family of submanifolds such that $A$ and $B_s$ are always transverse. Is it true that the homotopy type of $A \cap B_s$ is independent of $s$ ?
2. $A_s := f_s^{-1}(0)$ is the zero set of a smooth function $f_s : M \to \mathbb{R}$, and the family $A_s$ is always transversal to a fixed submanifold $B$. Is it true that the homotopy type of $A_s \cap B$ is independent of $s$ ?

Any help will be appreciated. Thanks in advance.