Popular music often features a high degree of stable harmonic patterns, e.g. four-chord songs, backdoor-progressions, rhythm changes, etc.. This facilitates the establishment of stylistic idioms and recognizability, and the changing frequencies of such patterns are closely linked to style and genre: new patterns arise while others die out. Here, we employ content-based transmission models from cultural evolution research and compare a range of 20th-century popular music genres from different geographical and cultural contexts. Our research goes beyond the state-of-the-art that mostly focuses on monophonic forms of music (e.g. melodies, chants). Moreover, prior work on the cultural evolution of harmony only considers a small vocabulary of chords with a binary distance metric (same or different). Here, we introduce music-theoretically sensible notions of harmonic distance between chords, such as chord inversion, extension, and functional equivalence. This allows us to arrive at more fine-grained results regarding relative influences of different kinds of harmonic relations on diachronic changes. Inferring the substitution probabilities for different chord classes, our results indicate an increasing usage of functional categories as well as of voice-leading, whereas chord extensions remain relatively stable. Our study provides a principled model-based methodology for cross-cultural research on the evolution of harmony.