Simulations with a 3D general circulation model (GCM) suggest that one potential driver behind the observed radius inflation in hot Jupiters may be the downward advection of energy from the highly irradiated photosphere into the deeper layers. Here, we compare dynamical heat transport within the non-inflated hot Jupiter WASP-43b and the canonical inflated hot Jupiter HD 209458b, with similar effective temperatures. We investigate to what extent the radiatively driven heating and cooling in the photosphere (at pressures smaller than 1 bar) influence the deeper temperature profile (at pressures between 1 to 700 bar). Our simulations with the new non-gray 3D radiation-hydrodynamical model expeRT/MITgcm show that the deep temperature profile of WASP-43b is associated with a relatively cold adiabat. The deep layers of HD 209458b, however, do not converge and remain nearly unchanged regardless of whether a cold or a hot initial state is used. Furthermore, we show that different flow structures in the deep atmospheric layers arise. There, we find that WASP-43b exhibits a deep equatorial jet, driven by the relatively fast tidally locked rotation of this planet (0.81 days), as compared to HD 209458b (3.47 days). However, by comparing simulations with different rotation periods, we find that the resulting flow structures only marginally influence the temperature evolution in the deep atmosphere, which is almost completely dominated by radiative heating and cooling. Furthermore, we find that the evolution of deeper layers can influence the 3D temperature structure in the photosphere of WASP-43b. Thus, dayside emission spectra of WASP-43b may shed more light onto the dynamical processes occurring at greater depths.