Assume $A \subseteq C \subseteq B$ are integral domains, with $B$ flat over $A$.
Generally, $B$ is not necessarily flat over $C$.
For example, see van den Essen's book "Polynomial Automorphisms and the Jacobian Conjecture", page 294, Exercise 4: $k[x^2] \subset k[x^2,x^3] \subset k[x]$. $k[x]$ is a free $k[x^2]$-module of rank $2$ (hence flat). But $k[x]$ is NOT a flat $k[x^2,x^3]$-module.
My question: What additional conditions are needed for flatness of $B$ over $C$?
Please notice that the above question is not the same as Projectivity of $B$ over $C$, given $A \subset C \subset B$, since here the extensions need not be finitely generated as modules, while there the extensions are assumed to be finitely generated as modules. Also here there is no assumption on $C$ over $A$, while there $C$ is assumed to be a free $A$-module.
On the other hand, here there is an additional condition that all rings are assumed to be integral domains, while there possibly not.
EDIT: Given $C \subseteq B$, there exist some really nice results showing when $B$ is flat over $C$ (for example, by Nagata, Chase, Ohm and Rush, Ohi). However, I wish to know how the additional information I have (namely, $A \subseteq C \subseteq B$ with $B$ flat over $A$) helps to decide when $B$ is flat over $C$.