There is some result in the case of Lie groupoids and I believe this is related.

Given Lie groupoids $\mathcal{G},\mathcal{H}$ a morphism of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$ comes from what is called a $\mathcal{G}-\mathcal{H}$ bibundle $P$. This bibundle comes from a morphism of Lie groupoids $\mathcal{G}\rightarrow\mathcal{H}$ if and only if the anchor map $a:P\rightarrow \mathcal{G}_0$ has a global section.

This can be found in proposition $3.36$ of Orbifold as stacks. I think similar result in case of Algebriac geometry can be said.

There is some result in the case of Lie groupoids and I believe this is related.

Given Lie groupoids $\mathcal{G},\mathcal{H}$ a morphism of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$ comes from what is called a $\mathcal{G}-\mathcal{H}$ bibundle $P$. This bibundle comes from a morphism of Lie groupoids $\mathcal{G}\rightarrow\mathcal{H}$ if and only if the anchor map $a:P\rightarrow \mathcal{G}_0$ has a global section.

This can be found in proposition $3.36$ of Orbifold as stacks. I think similar result in case of Algebraic geometry can be said.