LUX is a proposed facility for ultrafast X-ray science, based on an electron beam accelerated to ~3-GeV energy in a superconducting, recirculating linac.Included in the design are multiple FEL beamlines which use the harmonic cascade approach to produce coherent XUV & soft X-ray emission beginning with a strong input external laser seed at ~200 nm wavelength. Each cascade module generally operates in the low-gain regime and is composed of a radiator together with a modulator section, separated by a magnetic chicane. The chicane temporally delays the electron beam pulse in order that a "virgin" pulse region (with undegraded energy spread) be brought into synchronism with the radiation pulse. For each cascade, the output photon energy can be selected over a wide range by varying the seed laser wavelength and the field strength in the undulators. We present numerical simulation results, as well as those from analytical models, to examine predicted FEL performance. We also discuss lattice considerations pertinent to harmonic cascade FELs, as well as sensitivity studies and requirements on the electron beam.