Is the pushforward of a flat modules flat

Question

Let $k$ be an algebraically closed field of characteristic zero. Let $X, Y, Z$ be $k$-schemes of finite type. Let $F$ be a coherent sheaf on $X \times Y \times Z$, flat over $Y \times Z$. Is $\mathrm{pr}_{23_*}F$ flat over $Z$, where $\mathrm{pr}_{23}:X \times Y \times Z \to Y \times Z$ is the natural projection map?