  We study dynamics of a mobile impurity moving in a one-dimensional quantum
liquid. Such an impurity induces a strong non-linear depletion of the liquid
around it. The dispersion relation of the combined object, called depleton, is
a periodic function of its momentum with the period 2\pi n, where n is the mean
density of the liquid. In the adiabatic approximation a constant external force
acting on the impurity leads to the Bloch oscillations of the impurity around a
fixed position. Dynamically such oscillations are accompanied by the radiation
of energy in the form of phonons. The ensuing energy loss results in the
uniform drift of the oscillation center. We derive exact results for the
radiation-induced mobility as well as the thermal friction force in terms of
the equilibrium dispersion relation of the dressed impurity (depleton). These
results show that there is a wide range of external forces where the (drifted)
Bloch oscillations exist and may be observed experimentally.
