  A Thouless pump is a slowly driven one-dimensional band insulator which pumps
charge at a quantised rate. Previous work showed that pumping persists in
weakly disordered chains, and separately in clean chains at finite drive
frequency. We study the interplay of disorder and finite frequency, and show
that the pump rate always decays to zero due to non-adiabatic transitions
between the instantaneous eigenstates. However, the decay is slow, occurring on
a time-scale that is exponentially large in the period of the drive. In the
adiabatic limit, the band gap in the instantaneous spectrum closes at a
critical disorder strength above which pumping ceases. We predict the scaling
of the pump rate around this transition from a model of scattering between rare
states near the band edges. Our predictions can be experimentally tested in
ultracold atomic and photonic platforms.
