We systematically derive the dissipationless quantum kinetic equation for a multiband free fermionic system with $U$(1) symmetry. Using the Moyal product formalism, we fully band diagonalize the dynamics. Expanding to the second order in gradients, which is beyond the semiclassical limit, we give a complete analysis of the band-resolved thermodynamics and transport properties, especially those arising from the quantum geometric tensor. We apply our framework to a Bloch band theory under electric fields near equilibrium and find the linear and nonlinear transport coefficients. We also obtain the dynamical density-density response functions in the metallic case, including quantum metric corrections. Our results and approach can be applied very generally to multiband problems even in situations with spatially varying Hamiltonians and distributions.

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• Anomalous Hall effect
• Quantum kinetic theory
• Topological materials
• Free-electron model