With the purpose to optimize the functional properties of Heusler alloys for their use in solid-state refrigeration, the characteristics of the martensitic and magnetic transitions undergone by Ni50Mn25−xGa25Cux (x = 3–11) alloys have been studied. The results reveal that, for a Cu content of x = 5.5–7.5, a magnetostructural transition between paramagnetic austenite and ferromagnetic martensite takes place. In such a case, magnetic field and stress act in the same sense, lowering the critical combined fields to induce the transformation; moreover, magnetocaloric and elastocaloric effects are both direct, suggesting the use of combined fields to improve the overall refrigeration capacity of the alloy. Within this range of compositions, the measured transformation entropy is increased owing to the magnetic contribution to entropy, showing a maximum at composition x = 6, in which the magnetization jump at the transformation is the largest of the set. At the same time, the temperature hysteresis of the transformation displays a minimum at x = 6, attributed to the optimal lattice compatibility between austenite and martensite. We show that, among this system, the optimal caloric performance is found for the x = 6 composition, which displays high isothermal entropy changes (−36 J·kg−1·K−1 under 5 T and −8.5 J·kg−1·K−1 under 50 MPa), suitable working temperature (300 K), and low thermal hysteresis (3 K).