ABSTRACT We investigate the kinematics of the molecular gas in a sample of seven edge-on (i > 60°) galaxies identified as hosting large-scale outflows of ionized gas, using ALMA CO(1–0) observations at ∼1 kpc resolution. We build on Hogarth et al., where we find that molecular gas is more centrally concentrated in galaxies which host winds than in control objects. We perform full three-dimensional kinematic modelling with multiple combinations of kinematic components, allowing us to infer whether these objects share any similarities in their molecular gas structure. We use modelling to pinpoint the kinematic centre of each galaxy, in order to interpret their minor- and major-axis position velocity diagrams (PVDs). From the PVDs, we find that the bulk of the molecular gas in our galaxies is dynamically cold, tracing the rotation curves predicted by our symmetric, rotation-dominated models, but with minor flux asymmetries. Most notably, we find evidence of radial gas motion in a subset of our objects, which demonstrate a characteristic ‘twisting’ in their minor-axis PVDs generally associated with gas flow along the plane of a galaxy. In our highest S/N object, we include bi-symmetric radial flow in our kinematic model, and find (via the Bayesian Information Criterion) that the presence of radial gas motion is strongly favoured. This may provide one mechanism by which molecular gas and star formation are centrally concentrated, enabling the launch of massive ionized gas winds. However, in the remainder of our sample, we do not observe evidence that gas is being driven radially, once again emphasizing the variety of physical processes that may be powering the outflows in these objects, as originally noted in H21.