Let $f : X\rightarrow Y$ be a dominant morphism between smooth projective varieties over an algebraically closed field $k$.

If $k$ has characteristic zero, then $f$ is generically smooth in the sense that there exists a non-empty open subset of $Y$ over which $f$ is smooth.

In general, if I assume furthermore that $f$ is proper and $f_*\mathcal O_X=\mathcal O_Y$, so that $f$ cannot factor through a purely inseparable morphism $Y'\rightarrow Y$, does generic smoothness still hold ?