We restrict ourselves to the category of smooth manifolds and smooth maps.
Suppose we have a pair of smooth maps $f:A \to X$ and $g:A \to Y$. A pushout is a pair of smooth maps $p:X \to Z$ and $q:Y \to Z$ satisfying $pf=qg$ and the following universal property:
($*$) For every pair of smooth maps $r:X \to W$ and $s:Y \to W$ satisfying $rf=sg$, there is a unique smooth map $u: Z \to W$ such that $up=r$ and $ur=s$.
Question: Are any (relatively convenient) conditions for the existence of pushout known?
In the case of pullback, it is known that we have a pullback if one of $f:X \to B$ and $g:Y \to B$ is a submersion (or $f$ and $g$ are transverse). I am asking about such a condition for pushout.