For any finite-dimensional representation $V$ of a compact quantum group acting freely on a unital $C^*$-algebra $A$, we can form an associated finitely-generated projective module $A_V$ over the fixed-point subalgebra $B$ for this action. The module $A_V$ is the section module of the associated noncommutative vector bundle. Given an equivariant $C^*$-homomorphism $f$ from $A$ to $A'$, we get the induced K-theory map $f*$ from $K_0(B)$ to $K_0(B')$, where $B'$ is the fixed-point subalgebra of $A'$. Using Chern-Galois theory, we show that $f*([A_V])=[A'_V]$. As an application, we combine this formula with higher-rank graph $C^*$-algebra technology and index pairing computations to prove that the noncommutative line bundles associated via the diagonal U(1)-action on the multi-pullback quantum odd-dimensional spheres are pairwise stably non-isomorphic. In particular, we conclude that the tautological line bundles over the multi-pullback quantum complex projective spaces are stably non-trivial. The same reasoning and conclusions hold for noncommutative line bundles associated to Vaksman-Soibelman quantum spheres. (Partly based on joint work with David Pask, Aidan Sims and Bartosz Zieliński.)