  In this article, we consider products of random walks on finite groups with
moderate growth and discuss their cutoffs in the total variation. Based on
several comparison techniques, we are able to identify the total variation
cutoff of discrete time lazy random walks with the Hellinger distance cutoff of
continuous time random walks. Along with the cutoff criterion for Laplace
transforms, we derive a series of equivalent conditions on the existence of
cutoffs, including the existence of pre-cutoffs, Peres' product condition and a
formula generated by the graph diameters. For illustration, we consider
products of Heisenberg groups and randomized products of finite cycles.
