if the base chage of all fibres of a morphism is faithfully flat, do we have flatness of the morphism?

Question

Let $f:X\rightarrow Y$ be a morphism of schemes over $S$ (with possibly Noetherian conditions all over the place).

For every point $s\in S$, the morphism base changed to the fiber $f_s:X_s\rightarrow Y_s$ is faithfully flat.

Do we have that $f$ is flat? or even faithfully flat?

Edit: what if moreover both $X$ and $Y$ are flat over $S$?