  We study the competition between field-induced transport and trapping in a
disordered medium by studying biased random walks on random combs and the
bond-diluted Bethe lattice above the percolation threshold. While it is known
that the drift velocity vanishes above a critical threshold, here our focus is
on fluctuations, characterized by the variance of the transit times. On the
random comb, the variance is calculated exactly for a given realization of
disorder using a 'forward transport' limit which prohibits backward movement
along the backbone but allows an arbitrary number of excursions into
random-length branches. The disorder-averaged variance diverges at an earlier
threshold of the bias, implying a regime of anomalous fluctuations, although
the velocity is nonzero. Our results are verified numerically using a Monte
Carlo procedure that is adapted to account for ultra-slow returns from long
branches. On the Bethe lattice, we derive an upper bound for the critical
threshold bias for anomalous fluctuations of the mean transit time averaged
over disorder realizations. Finally, as for the passage to the vanishing
velocity regime, it is shown that the transition to the anomalous fluctuation
regime can change from continuous to first order depending on the distribution
of branch lengths.
