  We discuss the low-temperature dynamics of magnetic flux lines in samples
with a family of parallel twin planes. A current applied along the twin planes
drives flux motion in the direction transverse to the planes and acts like an
electric field applied to {\it one-dimensional} carriers in disordered
semiconductors. As in flux arrays with columnar pins, there is a regime where
the dynamics is dominated by superkink excitations that correspond to Mott
variable range hopping (VRH) of carriers. In one dimension, however, rare
events, such as large regions void of twin planes, can impede VRH and dominate
transport in samples that are sufficiently long in the direction of flux
motion. In short samples rare regions can be responsible for mesoscopic
effects.
