This document provides the supplementary material including a comprehensive related work, the complete proofs and extended evaluation results that support the manuscript, "Unified Characterization and Precoding for Non-Stationary Channels", that was accepted for publication at IEEE International Conference on Communications (ICC) 2022. Equations (1)--(34) refer to the equations from the main manuscript, and the Theorem, Lemma and Corollaries correspond to those from the manuscript.