  We show stabilisation of solutions to the sixth-order convective
Cahn-Hilliard equation. {The problem} has the structure of a gradient flow
perturbed by a quadratic destabilising term with coefficient $\delta>0$.
Through application of an abstract result by Carvalho-Langa-Robinson we show
that for small $\delta$ the equation has the structure of gradient flow in a
weak sense. On the way we prove a kind of Liouville theorem for eternal
solutions to parabolic problems. Finally, the desired stabilisation follows
from a powerful theorem due to Hale-Raugel.
