The Heusler alloys CoFeRGa (R = Ti, V, Cr, Mn, Cu, and Nb) have similar chemical compositions, but exhibit remarkably distinct electronic structures, magnetism and transport properties. These structures cover an extensive range of spin gapless semiconductors, half-metals, semiconductors and metals with either ferromagnetic, ferrimagnetic, antiferromagnetic, or nonmagnetic states. The Heusler alloys have three types of structures, namely, type-I, type-II, and type-III. By means of first-principles calculation combined with the Boltzmann equation within the consideration of spin-freedom, we explore the transport feature of the most stable structure (type-I). In addition, we provide evidence that all the considered materials are mechanically and dynamically stable, possessing high strength and toughness to resist compression and tensile strain. Moreover, the distinct electronic (metallic, insulating, and half-metallic) properties and magnetic behaviors originate mainly from a cooperative electron transfer and electronic structures have been verified by our calculation. Finally, we found that the tunable electronic structure with varied atomic numbers has significant influence on the spin-Seebeck effect. Correspondingly, the calculated spin-Seebeck coefficient of CoFeCrGa is -60.29 μV K-1 at 300 K, which is larger than that of other quaternary Heusler compounds. Our results provide a band-engineering platform to design Heusler structures with different electronic behaviors in isomorphic compounds, which provide the way for accelerating the pre-screening of materials to advance and for using the quaternary Heusler compounds for potential applications in spin caloritronic devices.