Finding novel half-metallic materials (HMMs) with Heusler structure is a focus of current research. Therefore, in this work, we aim to predict new HMMs in some rare-earth-element-contained equiatomic quaternary Heusler (EQH) compounds by using the first-principle calculations. In detail, the structural, electronic, magnetic, mechanical and half-metallic properties of LuFeMnZ (Z = Al, Ga), YFeMnZ (Z = Al, Ga), LuCoCrZ (Z = Si, Ge) and LaCoCrAl compounds were investigated. The calculated results showed that all the above-mentioned compounds are HMMs with complete spin polarisation around the Fermi level. The half-metallic behaviour, the spin-flip gaps and the total and atomic magnetic moments of these compounds for different lattice constants were also studied in detail. LuCoCrSi and LuCoCrGe compounds showed the largest spin-flip gaps of 0.42 eV at its equilibrium lattice constant and 0.39 eV at its strained lattice constant, respectively. Importantly, HMMs with a unique zero gap in one spin channel have also drawn much research attention in recent years. In the current work, the highly dispersive zero-gap half-metallic properties of LuCoCrGe compound were observed, when its lattice constant was extended to 6.70 and 6.80 Å. Finally, the mechanical stability and the phase stability of the above-mentioned compounds were analysed in detail.