Low-mass planets are known to undergo Type I migration and this process must have played a key role during the evolution of planetary systems. Analytical formulae for the disc torque have been derived assuming that the planet evolves on a fixed circular orbit. However, recent work has shown that in isothermal discs, a migrating protoplanet may also experience dynamical corotation torques that scale with the planet drift rate. The aim of this study is to examine whether dynamical corotation torques can also affect the migration of low-mass planets in non-isothermal discs. We performed 2D radiative hydrodynamical simulations to examine the orbital evolution outcome of migrating protoplanets as a function of disc mass. We find that a protoplanet can enter a fast migration regime when it migrates in the direction set by the entropy-related horseshoe drag and when the Toomre stability parameter is less than a threshold value below which the horseshoe region contracts into a tadpole-like region. In that case, an underdense trapped region appears near the planet, with an entropy excess compared to the ambient disc. If the viscosity and thermal diffusivity are small enough so that the entropy excess is conserved during migration, the planet then experiences strong corotation torques arising from the material flowing across the planet orbit. During fast migration, we observe that a protoplanet can pass through the zero-torque line predicted by static torques. We also find that fast migration may help in disrupting the mean-motion resonances that are formed by convergent migration of embryos.