Context. Spatially resolved observations of the ionized and molecular gas are critical for understanding the physical processes that govern the interstellar medium (ISM) in galaxies. Aims. To study the morpho-kinematic properties of the ionized and molecular gas in three dusty starburst galaxies at $z = 0.12-0.17$ to explore the relation between molecular ISM gas phase dynamics and the star-formation activity. Methods. We analyse $sim$kpc-scale ALMA CO(1--0) and seeing limited SINFONI Paschen-$alpha$ observations. We use a dynamical mass model, which accounts for beam-smearing effects, to constrain the CO-to-H$_2$ conversion factor. Results. One starburst galaxy shows irregular morphology which may indicate a major merger, while the other two systems show disc-like morpho-kinematics. The two disc-like starbursts show molecular gas velocity dispersion values comparable with that seen in local LIRG/ULIRGs, but in an ISM with molecular gas fraction and surface density values consistent to that reported for local star-forming galaxies. These molecular gas velocity dispersion values can be explained by assuming vertical pressure equilibrium. The star-formation activity is correlated with the molecular gas content suggesting depletion times of the order of $sim 0.1-1$ Gyr. The star formation rate surface density ($Sigma_{rm SFR}$) correlates with the ISM pressure set by self-gravity ($P_{rm grav}$) following a power law with an exponent close to 0.8. Conclusions. In dusty disc-like starburst galaxies, our data support the scenario in which the molecular gas velocity dispersion values are driven by the ISM pressure set by self-gravity, responsible to maintain the vertical pressure balance. The correlation between $Sigma_{rm SFR}$ and $P_{rm grav}$ suggests that, in these dusty starbursts galaxies, the star formation activity arises as a consequence of the ISM pressure balance.
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