Temperature-modulated differential scanning calorimetry of first-order transitions has led to many new observations. Some of these involve non-linear processes or deal with transformations of practically instantaneous response. The latter may cause serious lags within the calorimeter due to limited thermal conductivity of the sample and the instrument. In both cases the "reversing heat capacity" or a "complex heat capacity" is not a precise representation of the transition since both are computed from abbreviated Fourier transforms, limited to the evaluation of the first harmonic component. One has in these cases to work in the time-domain with the raw output. But even from these analyses in the time-domain many interesting new insights about the transition and the calorimeter performance can be generated.