Let $g\colon N\to M$ be an immersion.
Then, I think that $g^{-1}(p)$ is finite set or $0$-dimensional manifolds for all $p\in M$.
Now, let $g_t\colon N\to M$ be an one-parameter family of an immersion.
Let $F\colon N\times [0,1]\to M$ be the corresponding map given by $F(x,t)=g_t(x)$.
Is it true that $F^{-1}(p)$ is an one-dimensional manifold (possibly empty, of course)?