  We define an $\infty$-category of rational motives for cofiltered limits of
algebraic stacks, so-called pro-algebraic stacks. We show that it admits a
$6$-functor formalism for certain classes of morphisms. Our main example is the
stack of displays, closely related to the stack of Barsotti-Tate groups, seen
as a pro-algebraic stack via its truncations. On the way we deal with base
change along non-finite type morphisms such as non-finite Galois extensions.
