We have recently initiated the first spectroscopic dust reverberation programme on active galactic nuclei (AGN) in the near-infrared. Spectroscopy enables measurement of dust properties, such as flux, temperature and covering factor, with higher precision than photometry. In particular, it enables measurement of both luminosity-based dust radii and dust response times. Here we report results from a one-year campaign on NGC 5548. The hot dust responds to changes in the irradiating flux with a lag time of ~70 light-days, similar to what was previously found in photometric reverberation campaigns. The mean and rms spectra are similar, implying that the same dust component dominates both the emission and the variations. The dust lag time is consistent with the luminosity-based dust radius only if we assume a wavelength-independent dust emissivity-law, i.e. a blackbody, which is appropriate for grains of large sizes (of a few microns). For such grains the dust temperature is ~1450 K. Therefore, silicate grains have most likely evaporated and carbon is the main chemical component. But the hot dust is not close to its sublimation temperature, contrary to popular belief. This is further supported by our observation of temperature variations largely consistent with a heating/cooling process. Therefore, the inner dust-free region is enlarged and the dusty torus rather a "dusty wall", whose inner radius is expected to be luminosity-invariant. The dust-destruction mechanism that enlarges the dust-free region seems to partly affect also the dusty region. We observe a cyclical decrease in dust mass with implied dust reformation times of ~5-6 months.