In a material, how does the heat equation arise from phonons? And from electrons? What would be the starting point to derive the heat equation in a material? 
Generally, in insulators the heat is mediated via phonons while in metals (conductors), electrons are the main responsible for heat conduction. In doped semiconductors, the electronic and phononic contributions are about the same order of magnitude. In all cases, Fourier heat conduction applies.
So it should be possible to derive the heat equation starting from phonons (for insulators) and electrons (in metals). But how exactly would one accomplish this task?
Edit 1: I'm not really interested in the Drude's model. I would expect, at the very least, Bloch electrons or something more sophisticated.
 A: While I understand and validate your question, it does contain some unjustified assumptions. I'll try to give my best understanding and hope it clarifies, but while you dismiss the Drude-Sommerfeld model, the whole point of the renormalization group understanding of condensed matter is that the approximate explanations usually contain the correct and relevant phenomenological explanations.
1]For Conductors: the idea of Bloch theorem is to justify treating electrons as a free gas, as in, as long as there is few impurities/defects the Drude-Sommerfeld Model contains the correct physical explanations, the error being in O(1) factors. So, if you want to start with Bloch electrons and sparse impurities the first step is to evaluate the quantum mechanical  scattering cross section between electrons and nuclei. This is a textbook QM exercise, but you'll have to use the effective electron mass (material dependent) and the shielded coulomb potential (is just an exponential term on top of the coulomb interaction, but with scale factor material dependent). Once you have the cross section proceed as usual for Drude-Sommerfeld Model as in any textbook. The quantum mechanical treatment will only correct O(1) factors relating to the physical couplings, but otherwise the semiclassical model is entirely correct.
This assumes Born-Oppenheimer approximation. If you start working with this you'll trade wobbling defects/impurities by static ones plus phonons. In this case use Feynman rules to calculate the new cross section taking phonon-electron interactions and figure out the correction. Per usual of perturbation theory this will just give extra terms in the cross section.
Therefore the Drude-Sommerfeld Model already contains the physical explanation to heat transport in metals, all quantum mechanical considerations will only change numerical details on the scattering cross section, and thus will change only the numerical value of the heat conductivity. One need to worry only in case you have lots and lots of impurities/defects then you might end up with a Anderson Insulator, but then Bloch's Theorem no longer applies. Or strong interactions and then you might have Mott insulators. In any case go to [2]
2]For Insulators: Here, even simpler, electrons are bound to nuclei, so we only look for phonon propagation. This is a free boson gas, with corrections due to phonon-phonon interactions that arise in perturbation theory. 
So you can start with the "true hamiltonian" of a bunch of atoms interacting via coulomb forces, apply Born-Oppenheimer approximation and then you're left with bosons and fermion interacting perturbatively. Work the relevant cross section to desired coupling order, and then follow Drude-Sommerfeld cues, which are just an application Boltzmann equation.
If something is wrong it is because you have strong interactions, in this case the perturbation theory is not justified, and you end up with Anderson Effect or Mott Insulators. 
Also keep in mind that the Fourier equation is only experimentally valid for solid materials under small temperature differences. So a 10cm iron bar where each end is under hugely different temperatures will not obey fourier exchange, but will also have a wave-like transport (due to Bloch's electrons) that may dominate. Joel Lebowitz had a couple of papers in the late 1990s and early 2000s discussing the experimental limitations of the Fourier Equation.
