  We study operators of multiplication by $z^k$ in Dirichlet-type spaces
$D_\alpha$. We establish the existence of $k$ and $\alpha$ for which some
$z^k$-invariant subspaces of $D_\alpha$ do not satisfy the wandering property.
As a consequence of the proof, any Dirichlet-type space accepts an equivalent
norm under which the wandering property fails for some space for the operator
of multiplication by $z^k$, for any $k \geq 6$.
