We study a sample of 20 galaxies with an observed central drop in the stellar velocity dispersion (σ-drop) matched with a control sample of galaxies without a σ-drop. We search for correlations between σ-drops and the properties, primarily morphological, of the inner zones and discs of their host galaxies. Both morphological parameters and luminosity profiles are used to classify the samples. We find a larger fraction of Hα rings and nuclear dust spirals in the σ-drop sample. We also find that the fraction of Seyfert galaxies in the σ-drop sample is bigger than that of LINERs and that the reverse is true for the control sample. Our findings indicate that a σ-drop is very probably due to inflow-induced star formation in a dynamically cool disc, or in a gas ring, shock focused by an inner Lindblad resonance above a certain critical density level. The same mechanism that feeds the nuclear ring or the nuclear disc may well be responsible for the higher rate of Seyfert galaxies among the σ-drop hosts.