The existence of partially ionized, diffuse gas and dust clouds at kiloparsec scale distances above the central planes of edge-on, galaxy discs was an unexpected discovery about 20 years ago. Subsequent observations showed that this extended or extraplanar diffuse interstellar gas (EDIG) has rotation velocities approximately 10–20 per cent lower than those in the central plane, and has been hard to account for. Here, we present results of hydrodynamic models, with radiative cooling and heating from star formation. We find that in models with star formation generated stochastically across the disc, an extraplanar gas layer is generated as long as the star formation is sufficiently strong. However, this gas rotates at nearly the same speed as the midplane gas. We then studied a range of models with imposed spiral or bar waves in the disc. EDIG layers were also generated in these models, but primarily over the wave regions, not over the entire disc. Because of this partial coverage, the EDIG clouds move radially, as well as vertically, with the result that observed kinematic anomalies are reproduced. The implication is that the kinematic anomalies are the result of three-dimensional motions when the cylindrical symmetry of the disc is broken. Thus, the kinematic anomalies are the result of bars or strong waves, and more face-on galaxies with such waves should have an asymmetric EDIG component. The models also indicate that the EDIG can contain a significant fraction of cool gas, and that some star formation can be triggered at considerable heights above the disc mid-plane. We expect all of these effects to be more prominent in young, forming discs, to play a role in rapidly smoothing disc asymmetries and in working to self-regulate disc structure.