Rational motives on pro-algebraic stacks

We define an ∞-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. We deal with three examples that are included in this formalism. The stack of Barsotti-Tate groups, seen as a pro-algebraic stack via its truncations, the base change of an algebraic stack X over a field k to a Galois extension L, and a pro-algebraic stack related to Edidin-Graham's construction of equivariant Chow groups. We show that in the first example, the motive of the stack of Barsotti-Tate groups is Artin-Tate over Fₚ. For this, we show that motives on the stack of Barsotti-Tate groups are equivalent to motives on the stack of displays. In the second example, we naturally get an action of Gal(L/k) on the motive of XL and taking fixed points with respect to the continuous action yields the motive of X. In the last example, we recover the definitions of Hoskins-Pepin Lehalleur and Totaro.