  Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote
the support of the rounddown of D, and K the canonical divisor of X. We show
that any smooth family of canonically polarized varieties over X\S is
isotrivial if the divisor -(K+D) is ample. This result extends results of
Viehweg-Zuo and Kebekus-Kovacs.
  To prove this result we show that any extremal ray of the moving cone is
generated by a family of curves, and these curves are contracted after a
certain run of the minimal model program. In the log Fano case, this
generalizes a theorem by Araujo from the klt to the dlt case.
  In order to run the minimal model program, we have to switch to a
Q-factorialization of X. As Q-factorializations are generally not unique, we
use flops to pass from one Q-factorialization to another, proving the existence
of a Q-factorialization suitable for our purposes.
