Abstract A systematic study of the magnetic properties and both the temperature- and magnetic-field-induced structural transformations in the Ni50−xCoxMn39Sn11 (0⩽x⩽10) multifunctional alloys over a large temperature range from 500K down to 10K was performed. It is revealed that, with increasing x, the martensitic transformation temperatures first decrease slowly when 0⩽x⩽4 and then decrease rapidly when 5⩽x⩽8; no martensitic transformation was observed in the alloys with 9⩽x⩽10. The magnetic properties of these alloys are very sensitive to their chemical composition. The austenite in the alloys with 0⩽x⩽4 shows paramagnetic behavior and the martensite exhibits paramagnetic, superparamagnetic (SPM), and superspin glass (SSG) behaviors in different temperature ranges during cooling. In contrast, the austenite in the alloys with 5⩽x⩽8 is paramagnetic above its Curie temperature TC and ferromagnetic below TC, and the martensite shows SPM and SSG behaviors in different temperature ranges. The austenite in the alloys with 9⩽x⩽10, which remains stable down to 10K, shows paramagnetic and ferromagnetic behaviors above and below TC, respectively. The complete phase diagram for the Ni50−xCoxMn39Sn11 (0⩽x⩽10) alloy system, from high temperature down to 10K, is established. A significant magnetic-field-induced decrease of martensitic transformation temperatures and almost fully reversible magnetic-field-induced structural transformation are achieved across a broad temperature range in the alloys with 5⩽x⩽8. These results are important for understanding the composition and temperature-dependent functional properties, as well as their underlying mechanisms, in the Ni–(Co)–Mn–X (X=In, Sn, Sb) multifunctional alloys.