Let $f$ be a morphism of algebraic stacks which is representable by algebraic spaces. In Properties of Stacks, Section 100.3 we have defined what it means for $f$ to be universally submersive. Here is another characterization.
Lemma 101.12.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent
$f$ is universally submersive (as in Properties of Stacks, Section 100.3), and
for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the morphism of topological spaces $|\mathcal{Z} \times _\mathcal {Y} \mathcal{X}| \to |\mathcal{Z}|$ is submersive.
Proof. Assume (1), and let $\mathcal{Z} \to \mathcal{Y}$ be as in (2). Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Z}$. By assumption the morphism $V \times _\mathcal {Y} \mathcal{X} \to V$ of algebraic spaces is universally submersive, in particular the map $|V \times _\mathcal {Y} \mathcal{X}| \to |V|$ is submersive. By Properties of Stacks, Section 100.4 in the commutative diagram
the horizontal arrows are open and surjective, and moreover
is surjective. Hence as the left vertical arrow is submersive it follows that the right vertical arrow is submersive. This proves (2). The implication (2) $\Rightarrow $ (1) follows from the definitions. $\square$
Thus we may use the following natural definition.
Definition 101.12.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
We say $f$ is submersive1 if the continuous map $|\mathcal{X}| \to |\mathcal{Y}|$ is submersive, see Topology, Definition 5.6.3.
We say $f$ is universally submersive if for every morphism of algebraic stacks $\mathcal{Y}' \to \mathcal{Y}$ the base change $\mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ is submersive.
We note that a submersive morphism is in particular surjective.
Lemma 101.12.3. The base change of a universally submersive morphism of algebraic stacks by any morphism of algebraic stacks is universally submersive.
Proof. This is immediate from the definition. $\square$
Lemma 101.12.4. The composition of a pair of (universally) submersive morphisms of algebraic stacks is (universally) submersive.
Proof. Omitted. $\square$
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