  Neutron star electromagnetic activity produces pairs that fill their
magnetosphere represented to the zeroth order by the force-free approximation.
Neither dissipation nor acceleration nor radiation from charged particles is
expected from this simplified model. So far, only centred dipole magnetic
fields have been studied in this limit. In this paper, we explore the
consequences of a rotating off-centred dipole on the force-free magnetosphere,
showing the new magnetic field geometry, its spin-down luminosity as well as
the electromagnetic kick and torque felt by the neutron star. Solutions are
obtained by time-dependent numerical simulations of the force-free regime using
our pseudo-spectral code written in spherical coordinates. Our results are also
compared to known analytical expressions found for the off-centred vacuum
dipole by an expansion to lowest order in the parameter $\epsilon = d/R$, where
$d$ is the displacement of the dipole from the stellar centre and $R$ the
neutron star radius. The presence of a force-free plasma enhances the spin-down
luminosity as well as the electromagnetic kick and torque with respect to a
centred force-free dipole. The impact on isolated and binary neutron stars is
revised in light of these new results.
