Number of irreducible and connected components constant in flat families 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?
 A: A) You can easily reduce to the case that $S$ is the spectrum of a DVR $R$. Furthermore, by passing to a finite extension of $R$, you can assume that the components of the closed fiber are geometrically connected. Say that there are $d$ of them; then by semicontinuity the generic geometric fiber has at most $d$ components. Take a Stein factorization $X \to T \to S$; then $T$ has $d$ points over the closed point of $S$. Then $T$ is flat over $S$; this implies that the number of connected components of the generic geometric fiber of $X$ over $S$, that equals the degree of the generic fiber of $T$ over $S$, is at least $d$. This proves the equality.
B) The number of connected components of a geometric fiber is the dimension of H^0 of the structure sheaf of the fiber; the result follows from semicontinuity.
A: 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.

