Let $X$ be a scheme of finite type over a field and $f : X \rightarrow B$ and $g : X \rightarrow C$ two smooth surjective morphisms with connected fibres. I am wondering if the pushout of the diagram $$ \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{llllllllllll} X & \xrightarrow{f} & B \\ \downarrow{g} \\ C \\ \end{array} $$
exist. Intuitively, given two fibrations, I am looking to construct the largest possible common base.
There are plenty of references in the case where $f$ and $g$ are closed immersions. This question Pushout schemes/stacks is the closest I could get to, however, the counterexamples provided involve a finite morphism.
