  Let $L/\mathbb{Q}_p$ be a finite extension. We introduce $L$-typical prisms,
a mild generalization of prisms. Following ideas of Bhatt, Scholze, and Wu, we
show that certain vector bundles, called Laurent $F$-crystals, on the
$L$-typical prismatic site of a formal scheme $X$ over
$\mathrm{Spf}\mathcal{O}_L$ are equivalent to $\mathcal{O}_L$-linear local
systems on the generic fiber $X_\eta$. We also give comparison theorems for
computing the \'etale cohomology of a local system in terms of the cohomology
of its corresponding Laurent $F$-crystal. In the case $X =
\mathrm{Spf}\mathcal{O}_K$ for $K/L$ a $p$-adic field, we show that this
recovers the Kisin-Ren equivalence between Lubin-Tate
$(\varphi_q,\Gamma)$-modules and $\mathcal{O}_L$-linear representations of
$G_K$ and the results of Kupferer and Venjakob for computing Galois cohomology
in terms of Herr complexes of $(\varphi_q,\Gamma)$-modules. We can thus regard
Laurent $F$-crystals on the $L$-typical prismatic site as providing a suitable
notion of relative $(\varphi_q,\Gamma)$-modules.
