Tailed radio galaxies are shaped by ram pressure owing to the high-velocity motion of their host through the intracluster medium (ICM). Recent works have reported on the increasing complexity of the phenomenology of tailed galaxies, with departures from theoretical ageing models and evidence of re-energising mechanisms, which are yet unclear. The nearby (z=0.0894) galaxy cluster Abell 2142 hosts two tailed galaxies, namely T1 and T2, which exhibit peculiar morphological features. We aim to investigate the properties of T1 and T2 and constrain their spectral evolution, dynamics, and interactions with the ICM. We combined LOw Frequency Array (LOFAR), upgraded Giant Metrewave Radio Telescope (uGMRT), Very Large Array (VLA), and MeerKAT data (from 30MHz to 6.5GHz) to carry out a detailed spectral analysis of T1 and T2. We analysed surface brightness profiles, measured integrated and spatially-resolved spectral indices, and performed a comparison with single injection ageing models. Chandra X-ray data were used to search for discontinuities in the ICM properties in the direction of the targets. The spectral properties of T1 at low frequencies are predicted by ageing models, and provide constraints on the 3D dynamics of the host by assuming a constant velocity. However, sharp transitions along sub-regions of the tail, local surface brightness enhancements, and a spectral shape at high frequencies that is not predicted by models suggest a more complex scenario, possibly involving hydrodynamical instabilities and particle mixing. T2 exhibits unusual morphological and surface brightness features, and its spectral behaviour is not predicted by standard models. Two AGN outburst events during the infall of T2 towards the cluster centre could explain its properties.