  In the context of infrared subtraction algorithms beyond next-to-leading
order, it becomes necessary to consider multiple infrared limits of scattering
amplitudes, in which several particles become soft or collinear in a
strongly-ordered sequence. We study these limits from the point of view of
infrared factorisation, and we provide general definitions of strongly-ordered
soft and collinear kernels in terms of gauge-invariant operator matrix
elements. With these definitions in hand, it is possible to construct local
subtraction counterterms for strongly-ordered configurations. Because of their
factorised structure, these counterterms cancel infrared poles of real-virtual
contributions by construction. We test these ideas at tree level for multiple
emissions, and at one loop for single and double emissions, contributing to
NNLO and N3LO distributions, respectively.
