K-theory of regular compactification bundles

Let G be a split connected reductive algebraic group. Let (Formula presented.) be a (Formula presented.) -torsor over a smooth base scheme (Formula presented.) and X be a regular compactification of G. We describe the Grothendieck ring of the associated fibre bundle (Formula presented.), as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle over (Formula presented.). These are relative versions of the corresponding results on the Grothendieck ring of X in the case when (Formula presented.) is a point, and generalize the classical results on the Grothendieck rings of projective bundles, toric bundles and flag bundles.