I'm quite new to stacks, so this might be very easy. In particular, if there is a canonical reference I can consult for these things, please feel free to point it out.

Let $f:X\to Y$ be a $G$-equivariant morphism of schemes, for $G$ an affine group scheme (please feel free to relax the assumptions if it is natural, but this is the case I care about). Under what conditions on $f$ is the map between quotient stacks $[X/G]\to [Y/G]$ an open/closed immersion? Does it suffice for $f$ to be open/closed? In case not, does that become true after adding some reasonable assumptions on $G$?

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    $\begingroup$ At least if you assume that $G$ is flat over your base, that is always true (in the non-flat case, I am not immediately seeing whether these stacks are algebraic). A standard reference is Laumon and Moret-Bailly. Another reference is the Stacks Project. $\endgroup$ – Jason Starr Jun 2 '17 at 9:09

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