Medical applications of magnetocaloric effect (MCE) require possibility for precision shift of a temperature of the magnetic phase transition at the same MCE value and minimize irreversibility. Thus, detail dynamic MCE investigation of such alloys with non-toxic biocompatible dopants need to be done. In present work, the giant magnetocaloric effect, which is observed in the whole family of Fe-Rh alloys, has been investigated in Pd-doped samples in slowly cycled magnetic fields of up to 1.8 T in magnitude for a range of temperatures, 250 K < T < 350 K. The shift of the ferromagnetic/antiferromagnetic transition temperature down towards room temperature and the decrease in the MCE have been observed in these alloys in comparison with a quasi-equiatomic FeRh alloy. The measurements have also shown an asymmetric behaviour of the first order magnetic phase transition with respect to whether the transition is traversed by heating from lower temperatures or cooling from above. These peculiarities have been explained in the framework of the ab-initio density functional theory-based disordered local moment theory of the MCE. The results have been compared with the those for the non-doped FeRh alloy. Thus features of the first order magnetic phase transition that these alloys have in common have been revealed which enable some predictions to be made appropriate for practical applications.