  For a variety $X$ separated over a perfect field of characteristic $p>0$
which admits an embedding into a smooth variety, we establish an
anti-equivalence between the bounded derived categories of Cartier crystals on
$X$ and constructible $\mathbb Z/p \mathbb Z$-sheaves on the \'etale site
$X_{\text{\'et}}$. The key intermediate step is to extend the category of
locally finitely generated unit $\mathcal O_{F,X}$-modules for smooth schemes
introduced by Emerton and Kisin to embeddable schemes. On the one hand, this
category is equivalent to Cartier crystals. On the other hand, by using
Emerton-Kisin's Riemann-Hilbert correspondence, we show that it is equivalent
to Gabber's category of perverse sheaves in $D_c^b(X_{\text{\'et}},\mathbb Z/p
\mathbb Z)$.
