We use optical integral field spectroscopy from VLT/MUSE, as well as photometric observations from Hubble Space Telescope and VLT/HAWK-I, to study the morpho-kinematics of 17 low-mass (log(M/M⊙) < 9.5) MACS J0416.1-2403 cluster galaxies at R200 and five field galaxies with a redshift of z ∼ 0.4. By measuring fluxes of strong emission lines from the MUSE data, we have recovered the star formation rates, gas-phase metallicities, and spatially resolved gas kinematics, and we have also investigated the ionising mechanisms. We have analysed the structure and morphology of the galaxies from the optical and infrared photometric data, performing a multi-component decomposition into a bulge and a disk. The spatially resolved gas velocity fields of the cluster members and field galaxies were modelled using a 3D approach, which allowed us to retrieve their intrinsic gas kinematics, including the maximum rotation velocity and velocity dispersion. This enabled us to study scaling relations such as the Tully–Fisher and the stellar mass–S0.5 relation for low-mass galaxies in different environments and to search for signatures of cluster-specific processes using disturbed gas velocity fields as tracers. Most galaxies from our sample fall in the star-forming and composite region in the diagnostic diagram, which allows for the ionising sources in a galaxy to be disentangled. The cluster and field population can be classified as star-forming main-sequence galaxies, with only a sub-sample of four quenched systems. We observe significant scatter for the cluster galaxies in the mass-metallicity plane, and the lowest-mass systems deviate from the predictions of the fundamental metallicity relation, showing higher metallicities, whereas the higher-mass ones are in accordance with the model predictions. This might hint at the cutoff of pristine gas inflow and/or the removal of the hot halo gas as the mechanisms driving these offsets. Our morpho-kinematic analysis reveals a sub-sample of dwarfs with maximum velocities vmax < 50 km s−1 and vmax, gas/σgas < 1, which depart from the Tully–Fisher relation. This might indicate that their interstellar medium is affected by external environmental processes, such as ram pressure stripping. However, ∼30% of the cluster galaxies have rotation-dominated gas disks and follow the Tully–Fisher relation within 1σ. Using the S0.5 parameter, which links the dynamical support of ordered motions with that of random motions, we can differentiate between galaxies affected by gravitational processes and systems affected by hydrodynamical ones. In the stellar mass–S0.5 plane, both cluster and field galaxies follow a tight sequence, with only a sub-population of five galaxies strongly departing (> 4σ) from this relation, showing high σgas values. Both the morphology and kinematics of the outlier galaxies hint at a combination of pre-processing and cluster-specific interactions affecting their stellar and gas disks.