  In this article, we define the $m$-prismatic site and the $m$-$q$-crystalline
site, which is a higher level analogue of the prismatic site and the
$q$-crystalline site respectively. We prove a certain equivalence between the
category of crystals on the $m$-prismatic site (resp. the $m$-$q$-crystalline
site) and that on the prismatic site (resp. the $q$-crystalline site), which
can be regarded as the prismatic (resp. the $q$-crystalline) analogue of the
Frobenius descent due to Berthelot and the Cartier transform due to
Ogus-Vologodsky, Oyama and Xu. We also prove the equivalence between the
category of crystals on the $m$-prismatic site and that on the
$(m-1)$-$q$-crystalline site.
