LUX is a design study to develop concepts for future ultrafast x-ray facilities. Presently, LUX is based on an electron beam accelerated to ~;3-GeV energy in a superconducting, recirculating linac. Included in the design are multiple free-electron laser (FEL) beamlines which use the harmonic cascade approach to produce coherent XUV and soft X-ray emission beginning with a strong input seed at ~;200-nm wavelength obtained from a "conventional" laser. 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 a given 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 certain aspects of the predicted FEL performance. We also discuss lattice considerations pertinent to harmonic cascade FELs, some sensitivity studies and requirements on the undulator alignment, and temporal pulse evolution initiated by short input radiation seeds.