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)?
If $M$ and $N$ have the same dimension then $g$ being an immersion implies that $F$ is a submersion, so in this case yes.
Otherwise no. E.g. $N=\mathbb{R}$, $M=\mathbb{R}^2$, $g_t(x)=(x,t)$, $p=(0,1/2)$.