Is the reduction $X_{red}$ of a flat, finite, surjective scheme $X$ over an integral base $S$ still flat?

I could possibly add that I am already aware we can assume the base $S$ to be local and complete, and we can assume $X$ is local (and henselien). So far this hasn't seemed to help me.

My intuation is that I should be trying to show that the dimension of the residues of the sheaf of nilpotents $\mathcal{N}$ in the structure sheaf $\mathcal{O}_{X}$ of $X$ is constant over $S$.