  We will extend earlier transference results of Neuwirth and Ricard from the
context of noncommutative $L_p$-spaces associated with amenable groups to that
of noncommutative $L_p$-spaces over crossed products of amenable and
trace-preserving actions. Namely, if $T_m:L_p(\mathcal{L} G) \rightarrow
L_p(\mathcal{L} G)$ is a completely bounded Fourier multiplier, where
$\mathcal{L} G \subset \mathcal{B}(L_2 G)$ is the von Neumann algebra of $G$,
we will see that $\mathrm{Id} \rtimes T_m: L_p(\mathcal{M} \rtimes_\theta G)
\rightarrow L_p(\mathcal{M} \rtimes_\theta G)$ is also completely bounded and
that $$
  \| \mathrm{Id} \rtimes T_m: L_p(\mathcal{M} \rtimes_\theta G) \rightarrow
L_p(\mathcal{M} \rtimes_\theta G) \|_{\mathrm{cb}} \leq \| T_m \|_{\mathrm{cb}}
$$ provided that $\theta$ is amenable and trace-preserving. Furthermore, our
construction allow to extend $G$-equivariant completely bounded operators $S:
L_p(\mathcal{M}) \rightarrow L_p(\mathcal{M})$ to the crossed-product, so that
$$
  \|S \rtimes \mathrm{Id}: L_p(\mathcal{M} \rtimes_\theta G) \rightarrow
L_p(\mathcal{M} \rtimes_\theta G) \|_{\mathrm{cb}} \leq C^\frac{1}{p} \, \| S
\|_{\mathrm{cb}} $$ whenever $\theta$ is trace-preserving, amenable and its
generalized F{\o}lner sets satisfy certain accretivity property measured by the
constant $1 \leq C$. As a corollary, we will obtain stability results for
maximal $L_p$-bounds over crossed products. Such results imply the stability of
certain assumptions recently used to prove a noncommutative generalization of
the spectral H\"omander-Mikhlin theorem.
