We experimentally investigate the influence of varying pulse parameters on the spectral broadening, power spectral density, and relative intensity noise of mid-infrared (mid-IR) in-amplifier cascaded supercontinuum generation (SCG) by varying the pulse duration (35 ps, 1 ns, 3 ns) and repetition rate (100, 500, 1000 kHz). The system is characterized at the output of the erbium-ytterbium-doped in-amplifier SCG stage, the thulium/germanium power redistribution stage, and the passive ZBLAN fiber stage. In doing so, we demonstrate that the output of the later stages depends critically on the in-amplifier stage, and relate this to the onset of modulation instability.