  Large-scale vertical magnetic fields are believed to play a key role in the
evolution of protoplanetary discs. Associated with non-ideal effects, such as
ambipolar diffusion, they are known to launch a wind that could drive accretion
in the outer part of the disc ($R> 1$ AU). They also potentially lead to
self-organisation of the disc into large-scale axisymmetric structures, similar
to the rings recently imaged by sub-millimetre or near-infrared instruments
(ALMA and SPHERE). The aim of this paper is to investigate the mechanism behind
the formation of these gaseous rings, but also to understand the dust dynamics
and its emission in discs threaded by a large-scale magnetic field. To this
end, we performed global magneto-hydrodynamics (MHD) axisymmetric simulations
with ambipolar diffusion using a modified version of the PLUTO code. We
explored different magnetisations with the midplane $\beta$ parameter ranging
from $10^5$ to $10^3$ and included dust grains -- treated in the fluid
approximation -- ranging from $100 \mu$m to 1 cm in size. We first show that
the gaseous rings (associated with zonal flows) are tightly linked to the
existence of MHD winds. Secondly, we find that millimetre-size dust is highly
sedimented, with a typical scale height of 1 AU at $R=100$ AU for $\beta=10^4$,
compatible with recent ALMA observations. We also show that these grains
concentrate into pressure maxima associated with zonal flows, leading to the
formation of dusty rings. Using the radiative transfer code MCFOST, we computed
the dust emission and make predictions on the ring-gap contrast and the
spectral index that one might observe with interferometers like ALMA.
