A guiding problem in Kähler geometry is to find an equivalence between the existence of a given special metric and an algebro-geometric stability condition. In this context, a natural step towards proving the equivalence is to show that, given a manifold that admits a special metric, stable deformations of the manifold still admit a special metric. We propose a new technique to address this type of problem that relies on restricting to a finite-dimensional problem and applying the theory of moment maps and the moment map flow. We will show how to apply it to constant scalar curvature metrics and, if time permits, to holomorphic submersions.