The temperature profiles produced by various heating profiles are calculated from local heat transport models. The models take the heat flux to be the sum of heat diffusion and a non-diffusive heat flow, consistent with local measurements of heat transport. Two models are developed analytically in detail: (i) a heat pinch or excess temperature gradient model with constant coefficients; and (ii) a non-linear heat diffusion coefficient (χ) model. Both models predict weak (lesssim20%) temperature profile responses to physically relevant changes in the heat deposition profile – primarily because the temperature profile is a double integral of the heating profile. The model predictions are shown to agree with JET data for a variety of heating profiles ranging from peaked on-axis through approximately flat (NBI at high density) to localized off-axis (ICRH). The modest temperature profile responses that result from the models clarify why temperature profiles in many tokamaks are often characterized as exhibiting a high degree of 'profile consistency'. Global transport scaling laws are also derived from the two models. The non-linear model with χ ∝ dT/dr produces a non-linear energy confinement time (L-mode) scaling with input power, . The constant heat pinch or excess temperature gradient model leads to the offset linear law for the total stored energy W with Pin, W = τinc Pin + W(0), which describes JET auxiliary heating data quite well. It also provides definitions for the incremental energy confinement time , the heating effectiveness η, and the energy offset W(0). Considering both the temperature profile responses and the global transport scaling, the constant heat pinch or excess temperature gradient model is found to best characterize the present JET data. Finally, new methods are proposed for interpreting auxiliary heating data in terms of these local transport models.