  Under an oscillating mechanical drive, an amorphous material progressively
forgets its initial configuration and might eventually converge to a limit
cycle. Beyond quasistatic drivings, how structurally disordered systems lose or
record such memory remains theoretically challenging. Here we investigate these
issues in a minimal model system -- with quenched disorder and memory encoded
in a spatial pattern -- where the oscillating protocol can formally be replaced
by finite positive-velocity driving. We consider an elastic line driven at zero
temperature in a fixed disordered landscape, with bi-periodic boundary
conditions and tunable system size. This setting allows us to control the area
swept by the line at each cycle in a given disorder realisation, as would the
amplitude of an oscillating drive. We find that the line converges to
disorder-dependent limit cycles, jointly for its geometrical \emph{and}
velocity profiles. Moreover, the way it forgets its initial condition is
strongly coupled to the nature of the velocity dynamics it displays depending
on system size. We conclude on the implications of these results for the
response of amorphous materials under \emph{non}-quasistatic oscillating
protocols.
