TITLE: Number of irreducible and connected components constant in flat families
QUESTION [6 upvotes]: A) Let $f:F\rightarrow S$ be a flat proper morphism of schemes with geometrically normal fibers.  Then supposedly the number of $\textbf{connected}$ components of the geometric fibers is constant.  Why is this?  Without some kind of vanishing of cohomology or information on the base, I don't see why this is true.  
B) Furthermore, supposedly if $F$ is now a flat proper morphism with reduced, connected, nodal curves as geometric fibers, then there is a Zariski open subset of $S$ on which the fibers all have the same number of $\textbf{irreducible}$ components.  Why is this?
Finally, how far can these results be generalized?  For example, is B) true for any flat proper morphism?

REPLY [3 votes]: Part (A) is precisely the content of Theorem 4.7, (iii) in
Deligne, Pierre; Mumford, D., The irreducibility of the space of curves of a given genus, Publ. Math., Inst. Hautes Étud. Sci. 36, 75-109 (1969). ZBL0181.48803.