  We prove that crossed products of fiberwise essentially minimal
zero-dimensional dynamical systems have isomorphic $ K $-theory if and only if
the dynamical systems are strong orbit equivalent. Under the additional
assumption that the dynamical systems have no periodic points, this gives a
classification theorem including isomorphism of the $ C^* $-algebras as well.
We additionally explore the $ K $-theory of such crossed products and the
Bratteli diagrams associated to the dynamical systems.
