Active galactic nuclei (AGNs), star formation (SF), and galaxy interactions can drive turbulence in the gas of the interstellar medium (ISM), which, in turn, plays a role in SF taking place within galaxies. The impact on molecular gas is of particular importance, as it serves as the primary fuel for SF. Our goal is to investigate the origin of turbulence and the emission of molecular gas, as well as low-and-intermediate-ionisation gas, in the inner few kpc of both AGN hosts and star-forming galaxies (SFGs). We used archival JWST MIRI/MRS observations of a sample consisting of 54 galaxies at z < 0.1. We present flux measurements for the H2 S(5)λ6.9091 μm, [ArII]λ6.9853 μm, [FeII]λ5.3403 μm, and [ArIII]λ8.9914 μm emission lines along with velocity dispersion estimated by the W80 parameter. For galaxies with coronal line emission, we included measurements of the [MgV]λ5.6098 μm line. We compared the line ratios to photoionisation and shock models to explore the origin of the gas emission. AGNs exhibit broader emission lines than SFGs, with the largest velocity dispersions observed in radio-strong (RS) AGNs. The H2 gas is less turbulent compared to ionised gas, while coronal gas presents higher velocity dispersions. The W80 values for the ionised gas show a decrease when going from the nucleus out to radii of approximately 0.5─1 kpc, followed by an outward increase up to 2─3 kpc. In contrast, the H2 line widths generally display increasing profiles with distance from the center. Correlations between the W80 parameter and line ratios such as H2S(5)/[Ar II] and [Fe II]/[Ar II] indicate that the most turbulent gas is associated with shocks, enhancing H2 and [Fe II] emissions. Based on the observed line ratios and velocity dispersions, the [FeII] emission is consistent with predictions of fast shock models, while the H2 emission is likely associated with molecules formed in the post-shock region. We speculate that these shocked gas regions are produced by AGN outflows and jet-cloud interactions in AGN-dominated sources; whereas in SFGs, they might be created through stellar winds and mergers. This shock-induced gas heating may be an important mechanism of AGN (or stellar) feedback, preventing the gas from cooling and forming new stars.