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][1]. I think similar result in case of Algebraic geometry can be said.


  [1]: https://arxiv.org/pdf/0806.4160.pdf