CONDITIONING REGIMEN: Patients with no life-threatening viral or fungal infections within 1 month before the planned hematopoietic cell transplantation (HCT) receive alemtuzumab intravenously (IV) over 6 hours on day -10 and fludarabine phosphate IV over 30 minutes on days -4 to -2. They also undergo low-dose TBI on day 0. Patients with hemophagocytic lymphohistiocytosis (HLH), immune dysregulation polyendocrinopathy enteropathy X-linked (IPEX) syndrome, DiGeorge syndrome, or life-threatening viral or fungal infections within 1 month before the planned HCT receive fludarabine phosphate IV over 30 minutes on days -4 to -2 and undergo 2 low doses of TBI on day 0.