We have theoretically investigated the amplification dynamics in an Fe-doped chalcogenide chirped pulse amplifier based on an effective two-level model. The effects of the extraction-during-pumping technique, pump fluence and seed center wavelength on the output performances are thoroughly analyzed. The simulation results show that the gain narrowing and gain shifting effects are serious during the amplification process, while spectral pre-shaping of the seed pulses and the combination of different Fe-doped crystals can effectively suppress the gain narrowing effect. In particular, with appropriate crystal combination, ultra-broadband amplification can be realized, allowing for the generation of few-cycle mid-infrared laser pulses. The investigations conducted in this paper can provide a guideline for the design of Fe-doped chalcogenide chirped pulse amplifiers.