A reinforced concrete (RC) moment-resisting frame (MRF) with steel damper columns (SDCs) can be considered a damage-tolerant structure. SDCs behave as sacrificial members that absorb seismic energy prior to RC beams and columns. The behavior of such a structure depends on the strength balance of the RC MRF and SDCs, and the pinching behavior of RC members. In this article, the seismic behavior of an RC MRF with SDCs under pulse-like ground motion sequences is investigated by applying an extended incremental critical pseudo-multi-impulse analysis (ICPMIA). This article consists of two analytical studies. The first focuses on (a) the degradation in energy dissipation of an RC MRF with SDCs and (b) the increase in response period due to prior earthquake damage. An extended ICPMIA of RC MRF models is carried out. The second analytical study focuses on the influence of the pulse period of pulse-like ground motion sequences on the response of RC MRFs with SDCs. The main findings are as follows. (1) When the pulse velocities of the two multi impulses (MIs) are the same in sequential MIs, the peak displacement is larger than that of a single MI if the first and second MI have the same sign. This trend is notable when the SDC strength is relatively low, and the pinching behavior of RC beam is significant. (2) The degradation in energy dissipation of an RC MRF in the second input is notable when the pinching behavior of RC beams is significant and the SDC strength is relatively low, whereas such degradation is limited when the SDC strength is relatively high. (3) The increase in RC MRF response period in the second input is notable when the pinching behavior of RC beam is significant. (4) For nonlinear time history analysis (NTHA) using sequential ground pulses, the most critical period of the second pulse is longer than that of a single pulse. (5) The most critical response obtained from NTHA for the pulses in (4) can be approximated by the extended ICPMIA results.