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The next generation of space-based mm-wave telescopes, such as JAXA's LiteBIRD mission, require focal planes with thousands of detectors in order to achieve their science goals. Digital frequency-domain multiplexing (dfmux) techniques allow detector counts to scale without a linear growth in wire harnessing, sub-Kelvin refrigerator loads, and other scaling problems. In this paper, we describe the Digital Signal Processing (DSP) firmware executed in the design's FPGA. This firmware is responsible for synthesizing bias tones, performing dynamic feedback control of the bolometer voltage bias and/or Superconducting Quantum Interference Device (SQUID) nuller currents, demodulating and decimating bolometer channels into science data, and streaming the results for storage and eventual downlink. We describe how this firmware has been tailored for LiteBIRD, including the control path, improvements to power- and resource-efficiency, the addition of radiation-mitigation functions, and the integration of new bolometer biasing schemes that may help mitigate mission-specific design challenges. This paper is a companion piece to the description of the electronics platform in which the firmware operates.