Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties.

If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental group of $X$ in the topological fundamental group $\pi_1(B)$ of $B$ is of finite index.

What is $f$ is of relative dimension one? What are the properties of the morphism $f_*:\pi_1(X)\to \pi_1(B)$? Is it always surjective?

What if $f$ is of relative dimension $n$?

I am especially interested in the case where $f$ is non-isotrivial and $B$ is a curve.