  We consider a time-dependent extension of a perturbative mean-field approach
to the dirty boson problem by considering how switching on and off a weak
disorder potential affects the stationary state of an initially equilibrated
Bose-Einstein condensate by the emergence of a disorder-induced condensate
deformation. We find that in the switch on scenario the stationary condensate
deformation turns out to be a sum of an equilibrium part, that actually
corresponds to adiabatic switching on the disorder, and a dynamically-induced
part, where the latter depends on the particular driving protocol. If the
disorder is switched off afterwards, the resulting condensate deformation
acquires an additional dynamically-induced part in the long-time limit, while
the equilibrium part vanishes. We also present an appropriate generalization to
inhomogeneous trapped condensates. Our results demonstrate that the condensate
deformation represents an indicator of the generically non-equilibrium nature
of steady states of a Bose gas in a temporally controlled weak disorder.
