Given a homogeneous space X under a linear algebraic group G over the function field of an arithmetic curve, one can ask whether a local-global principle with respect to the patching setup introduced by Harbater, Hartmann and Krashen holds.

We will study this question by studying the associated quotient stack [X/G], which is a gerbe. This leads us to a discussion of local-global principles for gerbes and their relation to bitorsors.