Geometric flows and K\"ahler reduction

We investigate how to obtain various flows of K\"ahler metrics on a fixed manifold as variations of K\"ahler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchi's metric, the Calabi flow, the pseudo-Calabi flow of Chen-Zheng and the K\"ahler-Ricci flow. In the latter case we re-derive the V-soliton equation of La Nave-Tian.