  Many bundle gerbes constructed in practice are either infinite-dimensional,
or finite-dimensional but built using submersions that are far from being fibre
bundles. Murray and Stevenson proved that gerbes on simply-connected manifolds,
built from finite-dimensional fibre bundles with connected fibres, always have
a torsion $DD$-class. In this note I prove an analogous result for a wide class
of gerbes built from principal bundles, relaxing the requirements on the
fundamental group of the base and the connected components of the fibre,
allowing both to be nontrivial. This has consequences for possible models for
basic gerbes, the classification of crossed modules of finite-dimensional Lie
groups, the coefficient Lie-2-algebras for higher gauge theory on principal
2-bundles, and finite-dimensional twists of topological $K$-theory.
