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Context. We exploit the increased sensitivity of the recently installed adaptive optics SOUL at the LBT to obtain new high-spatialresolution near-infrared images of the massive young stellar object IRAS20126+4104 and its outflow. Aims. We aim to derive the jet proper motions and kinematics, as well as to study its photometric variability by combining the novel performances of SOUL together with previous near-infrared images. Methods. We used both broad-band (Ks, K') and narrow-band (Br H2) observations from a number of near-infrared cameras (UKIRT/UFTI, SUBARU/CIAO, TNG/NICS, LBT/PISCES, and LBT/LUCI1) to derive maps of the continuum and the H2 emission in the 2.12 µm line. Three sets of images, obtained with adaptive optics (AO) systems (CIAO, in 2003; FLAO, in 2012; SOUL, in 2020), allowed us to derive the proper motions of a large number of H2 knots along the jet. Photometry from all images was used to study the jet variability. Results. We derived knot proper motions in the range of 1.7-20.3 mas yr-1 (i.e. 13-158 km s-1 at 1.64 kpc), implying an average outflow tangential velocity of ~80 km s-1. The derived knot dynamical age spans a ~200-4000 yr interval. A ring-like H2 feature near the protostar location exhibits peculiar kinematics and may represent the outcome of a wide-angle wind impinging on the outflow cavity. Both H2 geometry and velocities agree with those inferred from proper motions of the H2O masers, located at a smaller distance from the protostar. Although the total H2 line emission from the knots does not exhibit time variations at a >~0.3 mag level, we have found a clear continuum flux variation (radiation scattered by the dust in the cavity opened by the jet) which is anti-correlated between the blue-shifted and red-shifted lobes and may be periodic (with a period of ~12-18 yr). We suggest that the continuum variability might be related to inner-disc oscillations which have also caused the jet precession. Conclusions. Our analysis shows that multi-epoch high-spatial-resolution imaging in the near-infrared is a powerful tool to unveil the physical properties of highly embedded massive protostars. [ABSTRACT FROM AUTHOR]