The classical reference by Mumford deals with GIT on schemes. Consider reductive groups only. Mumford first considered the theory for group actions on affine schemes, and then for a linearized action on a general scheme, he constructed the quotient by gluing affine quotients. Algebraic spaces are glued from affine schemes via étale morphisms rather than open immersions.
My question is can we mimic the GIT on schemes, and glue quotient affine schemes to obtain a quotient algebraic space. If we can, why is it not popular?
Quotients in the category of algebraic spaces are discussed in the following papers.
Kollár, János. Quotient spaces modulo algebraic groups. Ann. of Math. (2) 145 (1997), no. 1, 33--79.
Białynicki-Birula, A. (2002). Quotients by Actions of Groups. In: Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action. Encyclopaedia of Mathematical Sciences, vol 131. Springer, Berlin, Heidelberg.
Rydh, David. Existence and properties of geometric quotients. J. Algebraic Geom. 22 (2013), no. 4, 629--669.