Third-Party Natural Killer Cells and Mogamulizumab for the Treatment of Relapsed or Refractory Cutaneous T-cell Lymphomas or Adult T-Cell Leukemia/Lymphoma

Inclusion Criteria: * Able to understand and voluntarily sign an informed consent form * Age \>= 18 years at the time of signing the informed consent form * Able to adhere to the study visit schedule and other protocol requirements * Biopsy-proven, measurable, stage IB-IVB relapsed or refractory cutaneous T-cell lymphoma after 1 prior line of systemic therapy * Note: extracorporeal photopheresis will be considered a systemic therapy for this study * Patients with large cell transformation of cutaneous T cell lymphoma are eligible * Patients with adult T-cell leukemia/lymphoma (ATLL) of any stage and any subtypes. Patient must have had at least one standard chemotherapy and measurable disease at the time of enrollment * Patients who relapsed after autologous or allogeneic stem cell transplant are eligible * All cancer therapy, including radiation, topical steroid, and chemotherapy must have been discontinued at least 1 week or 3 half-lives whichever is the longest prior to treatment in this study. The only exceptions are participants who are symptomatic from their skin lesions and have been on corticosteroids for prolonged periods of time (\> 60 days) without change. These patients may continue use of either systemic steroids (equivalent to \< 10 mg per day of prednisone) or topical steroids if the frequency and dosage steroids has not changed for 21 days prior to the study. These participants should continue on the same dose of systemic/topical steroid throughout the study period unless they achieve a complete response at which time steroids can be tapered or discontinued. Patients are allowed to continue any medications with known activity in T cell lymphomas at the pre-enrollment doses for conditions other than T cell lymphomas (ie, steroids for sarcoidosis), as long as there is evidence of T cell lymphoma progression while patients were on these agents * Eastern Cooperative Oncology Group (ECOG) performance status of =\< 1 at study entry * Absolute neutrophil count \>= 1000/mm\^3 * Platelet count \>= 50,000/mm\^3 * Total bilirubin =\< 2 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and Alanine Aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 3 x ULN * AST (SGOT) and ALT (SGPT) =\< 5 x ULN in patients with documented hepatic involvement by lymphoma * Calculated creatinine clearance \>= 50 ml/min (by the Crockroft-Gault equation) * Disease free of prior malignancies for \>= 2 years with exception of currently treated basal cell, squamous cell carcinoma of the skin, or carcinoma in situ of the cervix or breast. Patients with early stage of prostate cancer under clinical surveillance without therapy are eligible. Patients with B-cell lymphomas treated with curative intent, and in remission for at least 2 years, may be in included (after discussion with principal investigator \[PI\]) * Negative serum pregnancy test at the time of enrollment for females of childbearing potential. Women who can get pregnant and men with partners who can become pregnant will be asked to practice a highly effective method of birth control while participating in the study that is considered medically acceptable by the study doctor. * Life expectancy \>= 90 days Exclusion Criteria: * Investigational therapies in the 2 weeks prior to beginning treatment on trial * Patients with active central nervous system (CNS) involvement with lymphoma * Patients with known human immunodeficiency virus (HIV) infection with CD4 \< 350 * Patients who had solid organ transplants * Evidence of active hepatitis B infection, based on positive surface antigen or hepatitis B deoxyribonucleic acid (DNA) polymerase chain reaction (PCR), or active hepatitis C infection based on positive PCR. Patients who are hepatitis B core antibody positive must take prophylaxis with lamivudine or equivalent and be willing to undergo monthly hepatitis B DNA PCR testing * Present or history of progressive multifocal leukoencephalopathy (PML) * Active grade II-IV acute or extensive chronic graft versus (vs.) host disease (GVHD) * Patients may take steroids at any dose for disease control up to 24 hours prior to study enrollment. Steroids must have been discontinued at least 1 week or 3 half-lives whichever is the longest prior to treatment in this study, per inclusion criteria above. Topical steroids are allowed for CTCL patients * Any illness, medical condition or organ system dysfunction which, in the investigator's opinion, could compromise the subject's safety * A cardiovascular disability status of New York Heart Association class \>= 2 * History of severe allergic reactions to humanized monoclonal antibodies * History of other malignancy that could affect compliance with the protocol or interpretation of results. Patients with a history of curatively treated basal or squamous cell carcinoma or Stage 1 melanoma of the skin or in situ carcinoma of the cervix are eligible. Patients with early stage of prostate cancer under clinical surveillance without therapy are eligible * Known hypersensitivity to any of the study drugs or analogs * Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment, or any major episode of infection requiring treatment with IV antibiotics or hospitalization (relating to the completion of the course of antibiotics) within 4 weeks prior study therapy * Clinically significant history of liver disease, including viral or other hepatitis, or cirrhosis * Receipt of live-virus vaccines within 28 days prior to the initiation of study treatment or need for live-virus vaccines at any time during study treatment * Recent major surgery (within 6 weeks prior to the start of study treatment) other than for diagnosis * Receiving immunosuppressive therapy * Prior therapy with mogamulizumab unless stopped previously for reasons other than progression or toxicity. * Pregnant or lactating, or intending to become pregnant during the study