  A slowly-varying or thin-layer multiscale assumption empowers macroscale
understanding of many physical scenarios from dispersion in pipes and rivers,
including beams, shells, and the modulation of nonlinear waves, to
homogenisation of micro-structures. Here we begin a new exploration of the
scenario where the given physics has non-local microscale interactions. We
rigorously analyse the dynamics of a basic example of shear dispersion. Near
each cross-section, the dynamics is expressed in the local moments of the
microscale non-local effects. Centre manifold theory then supports the local
modelling of the system's dynamics with coupling to neighbouring cross-sections
as a non-autonomous forcing. The union over all cross-sections then provides
powerful new support for the existence and emergence of a macroscale model
advection-diffusion PDE global in the large, finite-sized, domain. The approach
quantifies the accuracy of macroscale advection-diffusion approximations, and
has the potential to open previously intractable multiscale issues to new
insights.
