  We prove that the moduli b-divisor of an lc-trivial fibration from a log
canonical pair is log abundant. The result follows from a theorem on the
restriction of the moduli b-divisor, based on a theory of lc-trivial morphisms,
which allows us to treat $\mathbb{R}$-divisors and proper morphisms possibly
with disconnected fibres. We also prove a theorem on extending a finite cover
over a closed subvariety to that over a variety in arbitrary characteristic.
