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?



Are irreducible components of a flat family flat?
No to the first question. Let $Y$ be a nodal cubic curve and let $X$ be its connected two-sheeted covering space. Each of the two components of $X$ is the normalization of $Y$.


