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We show that if $f: X \to T$ is a surjective morphism between smoothprojective varieties over an algebraically closed field $k$ of characteristic$p>0$ with geometrically integral and non-uniruled generic fiber, then$K_{X/T}$ is pseudo-effective.  The hardest part of the proof is to show that there is a finite smoothnon-uniruled cover of the base, for which we show the following: if $T$ is asmooth projective variety over $k$, and $\mathcal{A}$ is an ample enough linebundle, then a cyclic cover of degree $p \nmid d$ given by a general element of$\left|\mathcal{A}^d\right|$ is not uniruled. For this we show the followingcohomological uniruledness condition, which might be of independent interest: asmooth projective variety $T$ of dimenion $n$ is not uniruled whenever thedimension of the semi-stable part of $H^n(T, \mathcal{O}_T)$ is greater thanthat of $H^{n-1}(T, \mathcal{O}_T)$.  Additionally we also show singular versions of all the above statements.

Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic