  We construct centrally large subalgebras in crossed products of $C (X, D)$ by
automorphisms in which $D$ is simple, $X$ is compact metrizable, the
automorphism induces a minimal homeomorphism of $X$, and a mild technical
assumption holds. We use this construction to prove structural properties of
the crossed product, such as (tracial) $Z$-stability, stable rank one, real
rank zero, and pure infiniteness, in a number of examples. Our examples are not
accessible via methods based on finite Rokhlin dimension, either because $D$ is
not $Z$-stable or because $X$ is infinite dimensional.
