We give a partial generalisation of Kobayashi-Hitchin correspondence where instead of holomorphic vector bundles, we consider arbitrary polarized fibrations. More precisely, for a polarized family of complex projective manifolds, we consider a version of Wess-Zumino-Witten equation (which can be seen as a generalisation of Hermite-Einstein equation) and show that if there is a solution to this equation, then the direct image sheaves associated with high tensor powers of the polarising line bundle have to be asymptotically semistable. This will be established by providing lower bounds on a fibered version of Yang-Mills functionals in terms of Harder-Narasimhan slopes of the direct images. We discuss the optimality of these lower bounds and, as an application, provide an analytic characterisation of a fibered version of generic nefness.
