  We define a new correspondence for pairs $(\mathbb{F},H)$ formed by a flag
manifold $\mathbb{F}$ together with an $H$-flux on $\mathbb{F}$. Given its role
within our correspondence, infinitesimal $T$-duality may be viewed as a source
of $H$-flux, in the sense that it contributes towards taking fluxless pairs
$(\mathbb{F},0)$ to pairs $(\mathbb{F}^\vee, H^\vee)$ carrying nontrivial flux
$H^\vee\neq 0$. We also illustrate how our correspondence exchanges complex
structures with symplectic ones up to $B$-transformations.
