According to the modern theory damage evolution under neutron irradiation depends on the fraction of self-interstitial atoms (SIAs) produced in the form of one-dimensionally glissile clusters. These clusters, having a low interaction cross-section with other defects, are absorbed mainly by grain boundaries and dislocations creating the so-called production bias. It is known empirically that addition of certain alloying elements influence many radiation effects, including swelling, however the mechanisms are unknown in many cases. In this paper we report the results of an extensive multi-technique atomistic level modeling study of SIA clusters mobility in bcc Fe-Ni alloys. We have found that Ni interacts strongly with the periphery of clusters affecting their mobility. The total effect is defined by the number of Ni atoms interacting with the cluster at the same time and can be significant even in low-Ni alloys. Thus 1nm (37SIAs) cluster is practically immobile at T<500K in the Fe-0.8at.% Ni alloy. Increasing cluster size and Ni content enhance cluster immobilization. This effect should have quite broad consequences in void swelling, matrix damage accumulation and radiation induced hardening and the results obtained help to better understand and predict the effects of radiation in Fe-Ni ferritic alloys.