This manuscript reports the theoretical results on the magneto-electronic properties of 8 new rare-earth-element-based equiatomic quaternary Heusler (EQH) half-metallic materials (HMMs) MCoTiZ (M = Lu, La; Z = Si, Ge, Sn) and ScCoTiN (N = Si, Ge) by means of the first-principles calculations. It is observed that all the EQH compounds (type III structure) studied in this work are stable in the ferromagnetic phase. The total magnetic moments of these EQH HMMs are all 2 μ B and they obey the well-known Slater-Pauling behavior, M t = Z t -18, M t is the total magnetic moment and Z t is the total number of valence electrons. The half-metallic band gaps (E HM ) of LuCoTiSi, LuCoTiGe, LuCoTiSn, LaCoTiSi, LaCoTiGe, LaCoTiSn, ScCoTiSi, and ScCoTiGe EQH compounds are respectively equal to 0.29 eV, 0.24 eV, 0.08 eV, 0.15 eV, 0.15 eV, 0.19 eV, 0.19 eV, and 0.15 eV. Compared to EQH HMMs only composed of 3d/4d transition metal elements, the E HM values of rare-earth-element-based EQH HMMs (except for LuCoTiSn) are larger, which is beneficial to the half-metallic stability in practical applications. Importantly, our work demonstrates that these compounds preserve their half-metallic states when the cubic primitive cell is tetragonally deformed or when the unit cell volume changes under the influence of uniform strain. Two types of strain are also applied to examine the magnetic and electronic properties of these compounds.