Flat algebras and tensor product

All rings are commutative. Suppose $B$ is a flat $A$-algebra, and that $M$ and $N$ are flat $B$-modules.

Is there a way to compare the two $A$-modules $M \otimes_A N$ and $M \otimes_B N$?


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If $B$ is a localization of $A$ then the natural map $M\otimes_A N \to M\otimes_B N$ will be an isomorphism (without any assumption on $M$ and $N$).

How definitive is this example?

Well, one way to think about your question is to take $M = N = B$. (You can’t get much flatter $B$-modules than this!) Then you are asking that the natural map $B\otimes_A B \to B$ be an isomorphism, which is to say that the diagonal map
$$\mathrm{Spec} \, B \hookrightarrow \mathrm{Spec \, B}\otimes_{\mathrm{Spec} \, A} \mathrm{Spec} \, B$$
be an isomorphism. This is asking that $A \to B$ be an epimorphism.

It is not so easy to find flat epimorphisms that are not localizations
(see the answers here, especially this one),
and so in practice I think that you should consider “$B$ is a localization of $A$” to be the most reasonable answer to the question of when this map is
an isomorphism.