What word can I use to describe time-varying spectral content?
The notions of harmonicity or non-stationary seem connoted.

Stochastic White noise or pink noise, or deterministic pure tones, are (strongly) structured in the spectral domain. So,  strongly structured, or strongly  spectrally-structured. So, in a soft manner, I would  describe the rich class as weakly structured, or weakly spectrally-structured.