The process of geometric quantization of a compact symplectic manifold depends on the choice of a complex structure, and a natural way to study this dependence is to consider the spaces of quantum states as a vector bundle over a space of complex structures. This idea is of particular interest in the context of moduli spaces of flat connections over a compact surface and the associated Verlinde bundle over Teichmüller space, for which there exists a canonical projectively flat connection. Parallel transport with respect to this connection allows one to define the Witten-Reshetikhin-Turaev invariant of mapping tori. In this talk, I will discuss the asymptotic expansion of this invariant as the level tends to infinity, and compute its first coefficient. This follows from a general semi-classical study of parallel transport in quantum bundles, via the theory of Berezin-Toeplitz quantization.