Let $k$ be a field (we can set it to be either perfect or algebraically closed if necessary), let $G$ be a (split) reductive group over $k$. Let $(X_i)$ be a filtered projective system of finite type $k$-schemes with affine transition maps and set $X = \lim_i X_i$ the projective limit of this system. We also assume $G$ acts on $X$ and the $X_i$ and that all the transition maps are $G$-equivariant.
Consider $[X/G]$ the quotient stack of $X$ by $G$ and all the quotient stacks $[X_i/G]$ we then have an inverse system in the $(2)$-category of algebraic stacks. Moreover we have a natural map $[X/G] \rightarrow \lim_i [X_i/G]$.
By the theory of Alper (see https://arxiv.org/abs/1005.2398) we can expect this map to be an adequate homeomorphism (loc cit) which is not far from being an isomorphism.
Question 1 : Is this map an isomorphism of algebraic stacks ?
The following paper suggests it is true (https://arxiv.org/abs/1307.4669) in the first line of the proof of 1.4.2.b.
Question 2 : Same as 1 but I only assume $k$ to be a noetherian ring ?
