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We extend the detailed study of the linearized dynamics obtained for cnoidal waves of the Korteweg–de Vries equation by Rodrigues [J. Funct. Anal. 274 (2018), pp. 2553–2605] to small-amplitude periodic traveling waves of the generalized Korteweg–de Vries equations that are not subject to Benjamin–Feir instability. With the adapted notion of stability, this provides for such waves, global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. When doing so, we actually prove that such results also hold for waves of arbitrary amplitude satisfying a form of spectral stability designated here as dispersive spectral stability. [ABSTRACT FROM AUTHOR]