Let $f : X \to Y$ be a flat proper morphism of complex varieties whose fibers are normal varieties. Is it true that $\mathrm{dim}_{\mathbb{Q}} H^i(X_t, \mathbb{Q})$ is constant?

For non-normal fibers, the cohomology rank can jump down. Can it also jump up?

If not, what is a counterexample? Is there any sort of semicontinuity (upper or lower) that does hold? I am particularly interested in the case $i = 1$.