Let's say I have a $U$-scheme $X$ and a group $U$-scheme $G$ acting on $X$. Then I can consider the quotient stack $[X/G]$ on $\text{Sch}/U$. The $T$-objects of this stack are simply $G$-torsors $P\to T$ equipped with a G-equivariant map $P\to X$. Morphisms are commutative squares compatible with the maps to $X$.
A quotient stack $[X/G]$ on $\text{Sch}/U$ is the same thing as a morphism $[X/G]\to U$ (with $U$ being considered as a stack). Now, if I take a scheme $V\to U$, I can take the pullback $[X/G]_V \to V$. This is an algebraic stack, and if one unravels the definition of the fibre product of stacks one sees that the $T$-objects of $[X/G]_V$ are as above, except $T\to U$ factors through $T\to S$.
It seems to me that $[X/G]_V \cong [X_V/G_V]$, where $X_V$ and $G_V$ are the pullbacks via $V\to U$ of $X$ and $G$ respectively. There is certainly a bijection between maps $P\to X$ and $P\to X_V$ when $P$ comes equipped with a map to $V$ (this is via $P\to T \to V$). I just don't know how to show $P\to X_V$ is $G_V$-equivariant when $P \to X$ is $G$-equivariant, or vice versa. Likewise, I don't know how to show that $P\to T$ (considered as $V$-schemes) are $G_V$-equivariant if and only if $P\to T$ (considered as $U$-schemes) are $G$-equivariant. I played around with some diagrams, but no luck.