  We consider the production of an arbitrary number of colour-singlet particles
near partonic threshold, and show that next-to-leading order cross sections for
this class of processes have a simple universal form at next-to-leading power
(NLP) in the energy of the emitted gluon radiation. Our analysis relies on a
recently derived factorisation formula for NLP threshold effects at amplitude
level, and therefore applies both if the leading-order process is tree-level
and if it is loop-induced. It holds for differential distributions as well. The
results can furthermore be seen as applications of recently derived
next-to-soft theorems for gauge theory amplitudes. We use our universal
expression to re-derive known results for the production of up to three Higgs
bosons at NLO in the large top mass limit, and for the hadro-production of a
pair of electroweak gauge bosons. Finally, we present new analytic results for
Higgs boson pair production at NLO and NLP, with exact top-mass dependence.
