Is a morphism between smooth varieties smooth if fibers are?
If you know that both $X$ and $Y$ are regular (even Cohen-Macauly), than constant fiber dimension implies flatness. See this for example. I believe this answers your question.
If you know that both $X$ and $Y$ are regular (even Cohen-Macauly), than constant fiber dimension implies flatness. See this for example. I believe this answers your question.