We present VLT-SINFONI K-band IFU spectroscopy of the central galaxies in the cool core clusters A1664, A2204 and PKS 0745-191, to probe the Pa-alpha and ro-vibrational H2 line emission. In A1664 the two emission-line velocity systems seen in our previous H-alpha data appear in both Pa-alpha and H2 emission, with notable morphological differences. The recession velocity of the red component of Pa-alpha increases linearly with decreasing radius, particularly along an 8 kpc filament aligned with the major axis of the underlying galaxy and the cluster X-ray emission. These kinematics are modelled as gravitational free-fall as gas cools rapidly out of the hot phase. In A2204 the gas shows 3 or 4 filaments reaching 10 kpc, three of which lie towards `ghost bubbles' seen in X-ray imaging. For PKS 0745-191, we confirm the twin-arm morphology of previous narrow-band images; the Pa-alpha kinematics suggest rotational motion about an axis aligned with the kpc-scale radio jet; on nucleus, we find a broad Pa-alpha component (FWHM 1700 km/s) and a secondary H2 system redshifted by +500 km/s. The H2 v=1-0 S(3)/Pa-alpha ratio is highest in isolated and extended regions where it matches the levels in the NGC 1275 filaments as modelled by Ferland et al. Regions with lower ratios highlight active star formation and are often kinematically quiescent (FWHM < 200 km/s). Our findings suggest that these clusters may be captured in different stages of the `cold feedback' cycle of Pizzolato & Soker, with A1664 in a brief phase of extreme cooling and star formation prior to an AGN heating event; PKS 0745-191 in outburst with the AGN accreting from a cool gas disk, and A2204 in a later phase where cool gas is dragged out of the galaxy by the buoyant rise of old radio bubbles (abridged).

16 pages, 16 figures, accepted by MNRAS