New results on electron heat wave propagation using ICRH power modulation in JET plasmas characterized by Internal Transport Barriers (ITB) are presented. The heat wave generated outside the ITB and travelling across it, always experiences a strong damping in the ITB layer, demonstrating a low level of transport and loss of stiffness. In some cases, however, the heat wave is strongly inflated in the region just outside the ITB, showing features of convective-like behaviour. In other cases, a second maximum in the perturbation amplitude is generated close to the ITB foot. Such peculiar types of behaviour can be explained on the basis of the existence of a critical temperature gradient length for the onset of turbulent transport. Convective-like features appear close to the threshold (i.e. just outside the ITB foot) when the value of the threshold is sufficiently high, with a good match with theoretical predictions of Trapped Electron Mode thresholds. The appearance of a second maximum is due to the oscillation of the temperature profile across the threshold in the case of a weak ITB. Simulations with an empirical critical gradient length model and with the theory based GLF23 model are presented. The difference with respect to previous results of cold pulse propagation across JET ITBs is also discussed.