  We show that the property of being rationally $K$-stable passes from the
fibers of a continuous $C(X)$-algebra to the ambient algebra, under the
assumption that the underlying space $X$ is compact, metrizable, and of finite
covering dimension. As an application, we show that a crossed product
C*-algebra is (rationally) $K$-stable provided the underlying C*-algebra is
(rationally) $K$-stable, and the action has finite Rokhlin dimension with
commuting towers.
