When flat base change is reduced

I am sorry, if this a very standard fact.

Let $f\colon X\to S$ be a morphism between varieties over field of characteristic zero. Let $\psi \colon S'\to S$ be a flat morphism. Is it true that $X\times_S S'$ is reduced?

If I need to assume something please suggest.(But in my case $f$ is not flat). On the other case, if you know something more general it also would be interesting.