In this note, we extend the detailed study of the linearized dynamics
obtained for cnoidal waves of the Korteweg--de Vries equation in \cite{JFA-R}
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.Comment: 15 page