  We study permanence properties of the classes of stable and so-called
D-stable C*-algebras, respectively. More precisely, we show that a
C_0(X)-algebra A is stable if all its fibres are, provided that the underlying
compact metrizable space X has finite covering dimension or that the Cuntz
semigroup of A is almost unperforated (a condition which is automatically
satisfied for C*-algebras absorbing the Jiang--Su algebra Z tensorially).
Furthermore, we prove that if D is a K_1-injective strongly self-absorbing
C*-algebra, then A absorbs D tensorially if and only if all its fibres do,
again provided that X is finite-dimensional. This latter statement generalizes
results of Blanchard and Kirchberg. We also show that the condition on the
dimension of X cannot be dropped. Along the way, we obtain a useful
characterization of when a C*-algebra with weakly unperforated Cuntz semigroup
is stable, which allows us to show that stability passes to extensions of
Z-absorbing C*-algebras.
