Definition I am aware of for Lie groupoid is that (among other things) the source and target maps $s,t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ are **submersions**.

On page 9 of Du Li's thesis *Higher Groupoid Actions, Bibundles,
and Differentiation* (arXiv:1512.04209) the author ask $s,t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ to be **surjective submersions**. This is the only place where I saw this kind of requirement.

What properties do we get easily assuming that source, target maps are surjective submersions rather than just submersions.

Lie groupoid $(M\rightrightarrows M)$ coming from Manifold $M$ is such that source, target maps are surjective submersions.

Lie groupoid $(G\rightrightarrows *)$ coming from a Lie group $G$ is such that source, target maps are surjective submersions.

Are there natural examples of Lie groupoids where source/target maps are **not** surjective submersions?

This condition continue on all results. For example, given a morphism of Lie groupids $\mathcal{G}\rightarrow\mathcal{H}$ and $\mathcal{K}\rightarrow \mathcal{H}$ the pullback $\mathcal{G}\times_{\mathcal{H}}\mathcal{K}$ is a Lie groupoid if $t\circ pr_2: \mathcal{G}_0\times_{\mathcal{H}_0}\mathcal{H}_1\rightarrow \mathcal{H}_0$ is a **sujective submersion**.