  We show that for any locally compact Hausdorff space $Y$ with finite covering
dimension and for any continuous flow $\mathbb{R} \curvearrowright Y$, the
resulting crossed product $C^*$-algebra $C_0(Y) \rtimes \mathbb{R}$ has finite
nuclear dimension. This generalizes previous results for free flows, where this
was proved using Rokhlin dimension techniques. As an application, we obtain
bounds for the nuclear dimension of $C^*$-algebras associated to
one-dimensional orientable foliations. This result is analogous to the one we
obtained earlier for non-free actions of $\mathbb{Z}$. Some novel techniques in
our proof include the use of a conditional expectation constructed from the
inclusion of a clopen subgroupoid, as well as the introduction of what we call
fiberwise groupoid coverings that help us build a link between foliation
$C^*$-algebras and crossed products.
