Let $f:X\to Y$ be a morphism of algebraic stacks.


If the **geometric** fibres of $f$ are algebraic spaces, then $f$ is representable by algebraic spaces.


I'm wondering about analogues of this fiberwise criterion in the context of gerbes and DM stacks.


For instance:


Q1. If the **geometric** fibres of $f$ are DM stacks, then is $f$ representable by DM stacks?


Q2. If there is a finite (abstract) group $G$ such that the **geometric** fibres of $f$ are $G$-gerbes, then is $f$ a $G$-gerbe?



Fiberwise criterion for a stack to be a gerbe
As for **Q2**, I don't think so. For example, consider $(BG \times (\mathbb{A}^1\setminus \{ 0 \}))\amalg BG\to \mathbb{A}^1$, everything over $\mathbb{C}$.


**Added** [Edit: this one doesn't work, see comments below]: for a flat, but perhaps more contrived, example, I think one can take $\mathscr{X}:=B\mu\_{3,\mathbb{Q}} \amalg B(\mathbb{Z}/3)\_{\mathbb{Q}}$ and $X:=\mathrm{Spec}(\mathbb{Q})\amalg\mathrm{Spec}(\mathbb{Q})$, where $\mathscr{X}\to X$ is the coarse moduli space map. Here $\mu\_{3,\mathbb{Q}}$ is the nonconstant group scheme over $\mathbb{Q}$ defined as a covariant functor on $\mathbb{Q}$-algebras by $A\mapsto$ third roots of $1$ in $A$, and $(\mathbb{Z}/3)\_{\mathbb{Q}}$ is the constant group scheme defined on $\mathbb{Q}$-algebras by $A\mapsto\mathbb{Z}/3$ (Edit: this is for connected $Spec(A)$, in general it's a *locally* constant group scheme, as by the way it has to be a locally constant sheaf of groups).
Now, I think the two basechanges along the two possible maps $\mathrm{Spec}(k)\to X$, $k$ algebraically closed, are both the gerbe $B\mu\_{3,k}=B(\mathbb{Z}/3)\_k$. 


