  We generalise partial results about the Yau-Tian-Donaldson correspondence on
ruled manifolds to bundles whose fibre is a classical flag variety. This is
done using Chern class computations involving the combinatorics of Schur
functors. The strongest results are obtained when working over a Riemann
surface. Weaker partial results are obtained for adiabatic polarisations over a
base of arbitrary dimension.
  We develop the notion of relative K-stability which embeds the idea of
working over a base variety into the theory of K-stability. Natural
constructions in filtered algebras equip the collection of test configuration
with extra structures. We illustrate these constructions with several examples.
