  The migration of a planet through a gaseous disc causes the locations of
their resonant interactions to drift and can alter the torques exerted between
the planet and the disc. We analyse the time-dependent dynamics of a
non-coorbital corotation resonance under these circumstances. The ratio of the
resonant torque in a steady state to the value given by Goldreich & Tremaine
(1979) depends essentially on two dimensionless quantities: a dimensionless
turbulent diffusion time-scale and a dimensionless radial drift speed. When the
drift speed is comparable to the libration speed and the viscosity is small,
the torque can become much larger than the unsaturated value in the absence of
migration, but is still proportional to the large-scale vortensity gradient in
the disc. Fluid that is trapped in the resonance and drifts with it acquires a
vortensity anomaly relative to its surroundings. If the anomaly is limited by
viscous diffusion in a steady state, the resulting torque is inversely
proportional to the viscosity, although a long time may be required to achieve
this state. A further, viscosity-independent, contribution to the torque comes
from fluid that streams through the resonant region. In other cases, torque
oscillations occur before the steady value is achieved. We discuss the
significance of these results for the evolution of eccentricity in
protoplanetary systems. We also describe the possible application of these
findings to the coorbital region and the concept of runaway (or type III)
migration. [Abridged]
