Consider a family of varieties f: X-> T, where T is a curve. We prove several inequalities about the slope of f, which are generalisations of the classical Xiao and Cornalba-Harris inequalities in the case where X is a surface. We then apply our results to the KSB moduli space of stable varieties to study the ample cone of such spaces. The talk is based on a joint work with Giulio Codogni and Filippo Viviani.