  The inclusive large-$p_T$ production of a single pion, jet or direct photon,
and Drell-Yan processes, are considered for proton-proton collisions in the
kinematical range expected for the fixed-target experiment AFTER, proposed at
LHC. For all these processes, predictions are given for the transverse
single-spin asymmetry, $A_N$, computed according to a Generalised Parton Model
previously discussed in the literature and based on TMD factorisation.
Comparisons with the results of a collinear twist-3 approach, recently
presented, are made and discussed.
