The ionisation cones of NGC 5728 have a deficit of molecular gas based on millimetre observations of CO (2-1) emission. Although photoionisation from the active nucleus may lead to suppression of this transition, warm molecular gas can still be present. We report the detection of eight mid-infrared rotational H2 lines throughout the central kiloparsec, including the ionisation cones, using integral field spectroscopic observations with JWST/MIRI MRS. The H2 line ratios, characteristic of a power-law temperature distribution, indicate that the gas is warmest where it enters the ionisation cone through disk rotation, suggestive of shock excitation. In the nucleus, where the data can be combined with an additional seven ro-vibrational H2 transitions, we find that moderate velocity (30 km s‑1) shocks in dense (105 cm‑3) gas, irradiated by an external UV field (G0 = 103), do provide a good match to the full set. The warm molecular gas in the ionisation cone that is traced by the H2 rotational lines has been heated to temperatures > 200 K. Outside of the ionisation cone the molecular gas kinematics are undisturbed. However, within the ionisation cone, the kinematics are substantially perturbed, indicative of a radial flow, but one that is quantitatively different from the ionised lines. We argue that this outflow is in the plane of the disk, implying a short 50 pc acceleration zone up to speeds of about 400 km s‑1 followed by an extended deceleration over ∼700 pc where it terminates. The deceleration is due to both the radially increasing galaxy mass, and mass-loading as ambient gas in the disk is swept up.