  In this paper we study families of projective manifolds with good minimal
models. After constructing a suitable moduli functor for polarized varieties
with canonical singularities, we show that, if not birationally isotrivial, the
base spaces of such families support subsheaves of log-pluridifferentials with
positive Kodaira dimension. Consequently we prove that, over special base
schemes, families of this type can only be birationally isotrivial and, as a
result, confirm a conjecture of Kebekus and Kov\'acs.
