  Perturbative cross-sections in QCD are beset by logarithms of kinematic
invariants, whose arguments vanish when heavy particles are produced near
threshold. Contributions of this type often need to be summed to all orders in
the coupling, in order to improve the behaviour of the perturbative expansion,
and it has long been known how to do this at leading power in the threshold
variable, using a variety of approaches. Recently, the problem of extending
this resummation to logarithms suppressed by a single power of the threshold
variable has received considerable attention. In this paper, we show that such
next-to-leading power (NLP) contributions can indeed be resummed, to leading
logarithmic (LL) accuracy, for any QCD process with a colour-singlet final
state, using a direct generalisation of the diagrammatic methods available at
leading power. We compare our results with other approaches, and comment on the
implications for further generalisations beyond leading-logarithmic accuracy.
