In their seminal paper The irreducibility of the space of curves of a given genus, Deligne and Mumford define a stable curve of genus $g$ over a scheme $S$ to be a flat, proper morphism $X\to S$, all of whose geometric fibres are stable curves of arithmetic genus $g$. I want to know is a stable curve over $S$ always locally of finite presentation? (This is true when $S$ is noetherian; I'm interested in the non-noetherian case.)
Some context:
The reason I ask is that the Stacks Project defines a family of nodal curves to be a flat, locally finitely presented morphism $X\to S$, all of whose fibres are nodal curves. I was curious why the assumption of local finite presentation seemed to be necessary in the definition of a family of nodal curves, but not in the definition of stable curves. Moreover, for families of nodal curves, pathologies can occur, such as a flat, proper morphism $X\to S$, all of whose fibres are nodal curves, but which is not locally finitely presented.
I should say that Deligne and Mumford assert that because $X\to S$ is flat and its geometric fibres are local complete intersections, then $X\to S$ is a locally complete intersection morphism, in particular is locally finitely presented. But I do not understand the argument.
