The temporal and spatial spectral evolution of the jets of active galactic nuclei (AGN) can be studied with multi-frequency, multi-epoch very-long-baselineinterferometry (VLBI) observations. The combination of both morphological (kinematical) and spectral parameters can be used to derive source-intrinsic physical properties, such as the magnetic field and the nonthermal particle density. Furthermore, we can trace the temporal variations in the source-intrinsic parameters during the flare, which may reflect the interaction between the underlying plasma and a traveling shock wave. In the first two papers of this series (Papers I and II), we analyzed the singledish light curves and the VLBI kinematics of the blazar CTA 102 and suggested a shock-shock interaction between a traveling and a standing shock wave as a possible scenario to explain the observed evolution of the component associated to the 2006 flare. In this paper we investigate the core shift and spectral evolution to test our hypothesis of a shock-shock interaction.