A part of Lemma 05AX in the Stacks Project states the following:
Let $k$ be a field. Let $f:X\to Y$ be a morphism of schemes locally of finite type over $k$. If $X$ is smooth over $k$ and $f$ is flat and surjective, then $Y$ is smooth over $k$.
Are there any similar situations where the smoothness of $X$ implies the smoothness of $Y$ with the flatness condition relaxed or replaced with a "more geometric" condition?
The reason why I ask such a question is that I do not find the flatness condition to be nearly as straightforward as the smoothness condition (this is only an opinion), yet it seems so necessary since examples like the one suggested by Mariano in the comments exist.