We calculate the response of a lead-based detector, such as the Helium and Lead Observatory (HALO) or its planned upgrade HALO-1kt to a galactic core-collapse supernova. We pay particular attention to the time dependence of the reaction rates. All reaction rates decrease as the neutrino luminosity exponentially drops during the cooling period but the ratio of one-neutron (1n) to two-neutron (2n) event rates in HALO is independent of this overall decrease. Nevertheless, we find that this ratio still changes with time due to the changing character of neutrino flavor transformations with the evolving conditions in the supernova. In the case of inverted hierarchy (IH), this is caused by the fact that the spectral splits become less and less sharp with the decreasing luminosity. In the case of normal hierarchy (NH), it is caused by the passage of the shock wave through the Mikheyev-Smirnov-Wolfenstein (MSW) resonance region. However, in both cases, we find that the change in the ratio of 1n to 2n event rates is limited to a few percent.