A study of the perturbation dynamics in a one-dimensional advective Cahn-Hilliard system, characterized by a nonvanishing driving force, is carried out to test the stability of a uniform basic solution. The linear stability of small-amplitude perturbations is analyzed both in the case of normal Fourier modes, with a given wave number, and in the case of wave packets localized in space. The dual nature of the instability, either of convective or absolute type, is studied, revealing that the driving force creates a gap between the parametric threshold to instability of normal modes and that to instability of wave packets. When the driving force is zero, also the gap between such thresholds disappears.