  We propose a simple but effective framework for producing examples of
covariant faithfully flat (generalised) Hopf-Galois extensions from a nested
pair of quantum homogeneous spaces. Our construction is modelled on the
classical situation of a homogeneous fibration $G/N \to G/M$, for $G$ a group,
and $N \subseteq M \subseteq G$ subgroups. Variations on Takeuchi's equivalence
and Schneider's descent theorem are presented in this context. Quantum flag
manifolds and their associated quantum Poisson homogeneous spaces are taken as
motivating examples. Moreover, a large collection of noncommutative fibrations
(in the spirit of Brzezi\'{n}ski and Szyma\'{n}ski) are constructed.
