Magnetic Heusler compounds (MHCs) have recently attracted great attention since these types of material provide novel functionalities in spintronic and magneto-electronic devices. Among the MHCs, some compounds have been predicted to be spin-filter semiconductors [also called magnetic semiconductors (MSs)], spin-gapless semiconductors (SGSs) or half-metals (HMs). In this work, by means of first-principles calculations, it is demonstrated that rare earth-based equiatomic quaternary Heusler (EQH) compounds with the formula MCoVZ (M = Lu, Y; Z = Si, Ge) are new spin-filter semiconductors with total magnetic moments of 3 mB. Furthermore, under uniform strain, there are physical transitions from spin-filter semiconductor (MS) ! SGS ! HM for EQH compounds with the formula LuCoVZ, and from HM!SGS!MS!SGS! HM for EQH compounds with the formula YCoVZ. Remarkably, for YCoVZ EQH compounds there are not only diverse physical transitions, but also different types of spin-gapless feature that can be observed with changing lattice constants. The structural stability of these four EQH compounds is also examined from the points of view of formation energy, cohesive energy and mechanical behaviour. This work is likely to inspire consideration of rare earthbased EQH compounds for application in future spintronic and magnetoelectronic devices.