Descend of morphisms of varieties

Given varieties $X,Y,W$ and dominant morphisms $\phi :X\to Y$ and $f:X\to W$ such that $f$ is constant on fibres of $\phi$ , we give sufficient conditions to guarantee that $f$ descends to a rational map or a morphism $Y\to W\mathrm{.}$ We pay special attention to the case that the ground field has positive characteristic. This extends previous works of Aichinger and Das, who proved similar results for some classes of affine varieties.