given a fibration of complex projective manifolds $f:x\rightarrow y$ with general fiber $f$, if the stable base locus of $-k_x $ is vertical then a theorem of chang establishes the inequality $\kappa(-k_x)\leq \kappa(-k_y) +\kappa(-k_f)$. in this talk i am going to discuss a generalization of this result to fibrations in positive characteristic satisfying certain tameness conditions. this is based on a joint project with marta benozzo and chi-kang chang.