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.
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