There exists a flat proper morphism f : X —> S all of whose geometric fibres are connected nodal curves such that f is not of finite presentation. An explicit example can be found in the examples chapter of the stacks project. Once you’ve understood why the example works, you easily see that you can even make an example where all the fibres are stable curves, S is connected, and the genus of the fibres jumps!
But let me go out on a limb here and make a wild guess: If you assume that there exists an integer g > 1 such that f is flat, proper, and all fibres are stable curves of genus g, then f is of finite presentation.
Why do I think this is true? I think it is true by analogy with the following results: A finite flat module need not be projective. A finite flat module over a local ring is free. Thus given a finite flat module over a scheme S then you get a well defined rank function. Then the module is finite locally free if and only if the rank function is locally constant in the Zariski toplogy (yet another characterization of finite projective modules, see Bourbaki, Commutative Algebra, Chapter II, Theorem 1).
I also think the following may be true: Given an integer d >= 0. If R —> A is a finite type, flat ring map all of whose geometric fibres are smooth and irreducible of dimension d, then R —> A is of finite presentation. (Irreducible implies nonempty. For this one I actually have a pretty good idea for how to prove it.)
Don’t do this at home kids!
What I mean by the last sentence is that, if you are doing actual moduli, you should just assume that X —> S is of finite presentation. In regards to this, note that if my wild guess is correct, then Definition 1.1 of Deligne-Mumford is the correct one. Thanks to Michael Thaddeus for pointing out that Deligne and Mumford only assume proper + flat + conditions on fibres.