We provide a detailed study of the dynamics obtained by linearizing the
Korteweg-de Vries equation about one of its periodic traveling waves, a cnoidal
wave. In a suitable sense, linearly analogous to space-modulated stability, we
prove global-in-time bounded stability in any Sobolev space, and asymptotic
stability of dispersive type. Furthermore, we provide both a leading-order
description of the dynamics in terms of slow modulation of local parameters and
asymptotic modulation systems and effective initial data for the evolution of
those parameters. This requires a global-in-time study of the dynamics
generated by a non normal operator with non constant coefficients. On the road
we also prove estimates on oscillatory integrals particularly suitable to
derive large-time asymptotic systems that could be of some general interest
