Turbulence propagation and temperature profile evolution are studied in heat flux-driven plasmas. A simple model consisting of coupled non-linear reaction-diffusion equations for both turbulence and heat transport is proposed to elucidate several aspects of apparent non-local profile dynamics. Self-consistent ExB shear feedback on turbulence intensity growth and transport is also included in the model. Temperature profile evolution is studied in the presence of an intensity pulse propagating inwards but also interacting with an outward propagating heat pulse. It is found that as the heat flux Q increases, the intensity pulse speed first grows as root Q and then decays as 1/Q, while the heat pulse speed finally saturates at the level given by neoclassical transport. Intensity pulse propagation can be effectively saturated at or above a critical heat flux, so that the formation of an internal transport barrier (ITB) can be triggered. This suggests that the ITB location is ultimately determined by both heat flux and edge turbulence conditions, and thus the ITB inhibits both the inward turbulence propagation and the outward turbulent heat transport. As a test of turbulence spreading dynamics, the intensity pulse propagation through gaps in turbulence excitation and its implications for profile response to off-axis heat deposition are also investigated. It is shown that the profile resilience phenomena can be recovered by taking into account intensity pulse propagation.