Let $f:X\rightarrow Y$ be a flat morphism of schemes of finite type over a field $k$, and assume $Y$ is irreducible. Let $X_1, \dots, X_n$ be the scheme-theoretic irreducible components of $X$ (i.e., including embedded components).

- Is it true that each $X_i$ is flat over $Y$?
- If there are counterexamples to flatness of the $X_i$, is it true at least that each of them has equidimensional fibers?