Metasurfaces allow for agile manipulation of incoming light using a single layer of resonators. Despite recent progress, it remains difficult to generate new spectral components using nonlinear surfaces, because of the limited interaction length of a pulse traveling through a single surface. Time-dependent surfaces offer an exciting alternative to overcome this limitation, but a self-consistent framework to describe this mechanism is lacking. Based on an analytical model and finite-difference time-domain numerical simulations, we obtain physical insight into the frequency-shifting process that occurs when broadband electromagnetic pulses interact with time-varying surfaces. In particular, we find that there is an intriguing relationship between the bandwidth of the incident pulse, the targeted frequency shift, and the number of Lorentzian resonators that need to be implemented. We also demonstrate that in certain parameter regimes, pulse distortion and a deviation of pulse amplitude cannot be avoided. These results are independent of the mechanism that generates the time dependence. They are also independent of the frequency, the geometry, size, and material of the unit cell, but they are rather a direct result of the subtle interplay between time-dependent Lorentzian resonances.