  We use the transverse K\"ahler-Ricci flow on the canonical foliation of a
closed Vaisman manifold to deform the Vaisman metric into another Vaisman
metric with a transverse K\"ahler-Einstein structure. We also study the main
features of such a manifold. Among other results, using techniques from the
theory of parabolic equations, we obtain a direct proof for the short time
existence of the solution for transverse {\K}-Ricci flow on Vaisman manifolds,
recovering in a particular setting a result of Bedulli, He and Vezzoni [J.
Geom. Anal. 28, 697--725 (2018)], but without employing the Molino structure
theorem. Moreover, we investigate Einstein-Weyl structures in the setting of
Vaisman manifolds and find their relationship with quasi-Einstein metrics. Some
examples are also provided to illustrate the main results.
