The distribution of warm molecular gas (1000-3000 K), traced by the near-IR H-2 2.12 mu m line, has been imaged with a resolution <0.5 arcsec in the central 1 kpc of seven nearby Seyfert galaxies. We find that this gas is highly concentrated towards the central 100 pc and that its morphology is often symmetrical. Lanes of warm H-2 gas are observed only in three cases (NGC 1068, NGC 1386 and Circinus) for which the morphology is much wider and extended than the dust filaments. We conclude that there is no one-to-one correlation between dust and warm gas. This indicates that, if the dust filaments and lanes of warm gas are radial streaming motions of fuelling material, they must represent two different phases of accretion: the dust filaments represent a colder phase than the gas close to the nucleus (within similar to 100 pc). We predict that the morphology of the nuclear dust at these scales should resemble that of the cold molecular gas (e.g. CO at 10-40 K), as we show for CenA and NGC 1566 by Atacama Large Millimeter/submillimeter Array (ALMA) observations, whereas the inner H2 gas traces a much warmer phase of material identified with warmer (40-500 K) molecular gas such as CO(6-5) or HCN (as shown by ALMA for NGC 1068 and NGC 1097). We also find that X-ray heating is the most likely dominant excitation mechanism of the H-2 gas for most sources.