We describe a dynamical state observed shortly above onset of the frozen wave instability. The transition to drifting waves, which are repeatedly created and destroyed, is a marked departure from the usual behavior of frozen waves, which are generally understood to remain motionless (on average) in the reference frame of the vibrating container. The spatial inhomogeneity of the underlying base flow, due both to the presence of the lateral walls and to the associated vibroequilibria effect, provides the driving mechanism. Energy arguments are used to understand the initial outward drift and the existence of a critical threshold which is estimated from the dependence of the drift velocity on the applied forcing. The dependence on container aspect ratio Γ is investigated, and drifting is seen to occur only when 1.5≲Γ≲3.5.