Aims. Gravitationally lensed systems allow a detailed view of galaxies at high redshift. High spatial- and spectral-resolution measurements of arc-like structures can offer unique constraints on the physical and dynamical properties of high-z systems. Methods. We present near-infrared spectra centred on the gravitational arcs of six known z ∼ 2 lensed star-forming galaxies of stellar masses of 10⁹⁻¹¹ M_⊙ and star formation rate (SFR) in the range between 10 and 400 M_⊙ yr⁻¹. Ground layer adaptive optics (AO)-assisted observations are obtained at the Large Binocular Telescope (LBT) with the LUCI spectrographs during the commissioning of the ARGOS facility. We used MOS masks with curved slits to follow the extended arched structures and study the diagnostic emission lines. LBT observations are used to demonstrate the spectroscopic capabilities of ARGOS. Results. Combining spatially resolved kinematic properties across the arc-like morphologies, emission line diagnostics and archival information, we distinguish between merging and rotationally supported systems, and reveal the possible presence of ejected gas. For galaxies that have evidence for outflows, we derive outflow energetics and mass-loading factors compatible with those observed for stellar winds in local and high-z galaxies. We also use flux ratio diagnostics to derive gas-phase metallicities. The low signal-to-noise ratio in the faint Hβ and nitrogen lines allows us to derive an upper limit of ≈0.15 dex for the spatial variations in metallicity along the slit for the lensed galaxy J1038. Conclusions. Analysed near-infrared spectra presented here represent the first scientific demonstration of performing AO-assisted multi-object spectroscopy with narrow curved-shape slits. The increased angular and spectral resolution, combined with the binocular operation mode with the 8.4 m wide eyes of LBT, will allow the characterisation of kinematic and chemical properties of a large sample of galaxies at high-z in the near future.