I am looking for some examples of gerbes over stacks (as defined in Understanding definition of gerbe over a stack) that comes from manifolds.

Let $M$ be a manifold then $\underline{M}$ is a stack associated to $M$ (I can give details if any one needs).

We know when to call a map of stacks $\mathcal{D}\rightarrow \mathcal{C}$ a gerbe over stack. Suppose $\mathcal{D}$ is coming from a manifold $M$ and $\mathcal{C}$ is coming from a manifold $N$ i.e., we have $\underline{M}\rightarrow \underline{N}$.

Given a map of stacks $F:\underline{M}\rightarrow \underline{N}$, we have unique map $F(id):M\rightarrow N$ associated to $F$.

What can we say about this map? Any comments are welcome.

First condition says that $F:\underline{M}\rightarrow \underline{N}$ is an epimorphism. Suppose $f:M\rightarrow N$ is a surjective submersion, then it follows that $F:\underline{M}\rightarrow \underline{N}$ is an epimorphism.

Second condition says that the diagonal map $\underline{M}\rightarrow \underline{M}\times_{\underline{N}}\underline{M}$ is an epimorphism. Suppose diagonal map $M\rightarrow M\times_N M$ is a surjective submersion, then $\underline{M}\rightarrow \underline{M}\times_{\underline{N}}\underline{M}$ is an epimorphism.

But then, $M\rightarrow M\times_N M$ is surjective implies that $f$ is injective.

An injective, surjectve submersion is a diffeomorphism. So, I landed up with a diffeomorphism conidtion on $M\rightarrow N$.

Do we have other non trivial examples?

Edit: Ycor has added algebraic geometry tag. I am also ok with algebraic geometry examples. I do not know much (not even the precise definition in that set up). Smooth manifolds are replaced by schemes (I guess). Given a scheme, there is an associated stack (algebraic stack??). Gerbe over stack in case of algebraic geometry of the form $\underline{X}\rightarrow \underline{Y}$ should correspond to some morphism of schemes $X\rightarrow Y$. It will be good if you can define gerbe over stack in algebraic geometry set up (loosely atleast) and the give examples.