  We prove that quasi-projective base spaces of smooth families of minimal
varieties of general type with maximal variation do not admit Zariski dense
entire curves. We deduce the fact that moduli stacks of polarized varieties of
this sort are Brody hyperbolic, answering a special case of a question of
Viehweg and Zuo. For two-dimensional bases, we show analogous results in the
more general case of families of varieties admitting a good minimal model.
