We report on the barocaloric and magnetocaloric effects in a series of low-hysteresis Ni-Mn-In magnetic shape memory alloys. We show that the behaviour exhibited by several quantities that characterise these caloric ef- fects (isothermal entropy change, adiabatic temperature change and refrig- erant capacity) can be rationalised in terms of the relative distance between the Curie point of the austenite and the martensitic transition temperature. It is found that the two caloric effects exhibit opposite trends. The be- haviour of the barocaloric effect parallels that exhibited by the transition entropy change, thereby showing larger values for weakly magnetic samples. Regarding the magnetocaloric effect, the entropy change is maximum for those samples transforming martensitically close to the Curie point of the austenite. Such a maximum value does not correspond to the maximum adi- abatic temperature change, and samples with martensitic transition slightly below the Curie point do have larger temperature changes as a result of the strongest sensitivity of the transition to the magnetic eld.