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 $G$?