Let $f:X\to B$ and $g:Y \to B$ be two morphisms of varieties over $\mathbb C$. If for every closed point $b\in B$, the fibers $f^{-1}(b)$ and $g^{-1}(b)$ are isomorphic. I want to know that if they are locally isomorphic? i.e., can we find an open set $U$ for every $b\in B$ such that $f|_{f^{-1}(U)}$ and $g|_{g^{-1}(U)}$ are isomorphic?
If not (which I tend to believe), under what conditions can it be true?