<p>The selection of a proper heat conduction model beyond Fourier for complex engineering problems is not straightforward, and the appearance of multiple time scales makes their treatment even more difficult. The present study offers a set of selection criteria for heat equations beyond Fourier. Based on these criteria, it highlights the Guyer-Krumhansl and Jeffreys heat equations as highly viable alternatives due to their flexible continuum thermodynamic background, the ability to inherit results from phonon hydrodynamics, and their applicability to strongly heterogeneous materials such as composite foams. Furthermore, we propose that any heat equation beyond Fourier should be defined by the related linear Onsagerian relations, which clarify the origin of the observable macroscopic transport coefficients. We study the consequences of temperature-dependent thermal conductivity and their relations to the static and dynamic heat conduction time scales, particularly in the case of the Jeffreys heat equation. We also provide numerical demonstrations to support our findings.</p>