  For every $d \geq 4$, we construct a $d$-dimensional, log canonical,
$K$-trivial variety with the property that two general fibers of its Albanese
morphism are not birational. This provides a strong counterexample to the
Beauville--Bogomolov decomposition in the log canonical setting. This
construction can also be adapted to construct a smooth quasi-projective variety
of logarithmic Kodaira dimension 0 whose quasi-Albanese morphism has maximal
variation. On the positive side, we show that the Albanese morphism for log
canonical pairs with nef anti-canonical class is a locally stable family of
pairs.
