Ni-Co-Mn-Sb-based Heusler shape-memory alloys that undergo a martensitic-structural transition around room temperature are well known for exhibiting large magnetic entropy change and elastocaloric effect. Here, we report the observation of a large adiabatic temperature change of -11K in a Ni-Co-Mn-Sb system by using direct adiabatic temperature-change measurements in pulsed magnetic fields. We show that a large magnetic cooling can be achieved in a wide temperature range spanning from 120 to 270 K by purposefully varying the chemical composition. The temperature- and field-dependent irreversibility of the effect is analyzed through a detailed experimental study of the protocol-dependent magnetocaloric effect. The present study is an important contribution towards the understanding of irreversible magnetocaloric effects in materials with magnetostructural transition.