CD8+ Memory T-Cells as Consolidative Therapy After Donor Non-myeloablative Hematopoietic Cell Transplant in Treating Patients With Leukemia or Lymphoma

Inclusion Criteria: * Must have a human leukocyte antigen (HLA)-matched or single allele-mismatched adult sibling serving as donor * Must have a myeloid or lymphoid malignant disease that is treated with TLI and ATG reduced intensity conditioning for allogeneic transplant (any of the following AML, myelodysplastic syndrome \[MDS\], myeloproliferative disease \[MPD\], CLL, B or T-cell NHL, HL) * Patients who due to age, pre-existing medical conditions, or, prior therapy are considered to be at high risk for regimen related toxicity associated with fully ablative transplant conditioning, and therefore reduced intensity conditioning is recommended * Ability to understand and the willingness to sign a written informed consent document; patients must have signed informed consent to participate in the trial * DONOR: Must be an HLA-matched or single allele mismatched sibling of enrolled transplant patient * DONOR: Must be 18-75 years of age, inclusive * DONOR: Must be in a state of general good health and have completed a donor evaluation with history, medical examination and standard blood tests within 35 days of starting the hematopoietic cell collection procedure; in order to fairly represent the interests of the donor, the donor evaluation and consent will be performed by a study team member other than the recipient's attending physician * DONOR: Must have a white blood cell count \> 3.5 x 10\^9/liter, platelets \> 150 x 10\^9/liter and hematocrit \> 35% * DONOR: Must be capable of undergoing leukapheresis * DONOR: Must be able to understand and sign informed consent * DONOR: Must not be seropositive for HIV 1 and 2, hepatitis B surface antigen, hepatitis C antibody, human T-lymphotropic virus (HTLV) antibody, cytomegalovirus (CMV) immunoglobulin M (IgM), or rapid plasma reagin (RPR) (Treponema); donors with prior evidence of hepatitis B core antibody positivity will have a polymerase chain reaction (PCR) test done to evaluate for hepatitis B infection; donors with a positive hepatitis B PCR test are excluded * DONOR: Females must not be pregnant or lactating * DONOR: Must not have psychological traits or psychological or medical conditions which make them unlikely to tolerate the procedure * DONOR: Must not have developed a new malignancy requiring chemotherapy or radiation in the interval since apheresis for initial hematocrit (HCT) * PATIENT CRITERIA FOR PROCEEDING WITH CD8+ MEMORY T-CELL INFUSION: * Patients must be beyond day 30 and before day 60 after transplant * Patients must have evidence of mixed CD3 T-cell chimerism based on the day +28 (+/- 7 days) blood sample showing \>= 5% and =\< 95% donor type cells * Patients must have no evidence of active graft-versus-host disease at the time of the CD8+ memory T-cell infusion; patients with a history of acute GVHD overall grade II based on skin only involvement or upper gastrointestinal (GI) tract involvement only will be eligible; patients with a history of liver or lower GI tract GVHD will not be eligible * Patients must be on single immune suppression therapy with either tacrolimus or cyclosporine at the time of CD8+ memory T-cell infusion; prednisone at a physiologic dose of 5 mg per day or less is allowed * Patients must have a Karnofsky performance status of \>= 60% at the time of the CD8+ memory T-cell infusion * Patients must not have an uncontrolled bacterial, fungal or viral infection, defined as progressive symptoms despite therapy, at the time of the CD8+ memory T-cell infusion; asymptomatic viremia is allowed * Patients must have adequate organ function and performance status at the time of the CD8+ memory T-cell infusion, defined by the following: * Total bilirubin =\< 4 mg/dL * SGOT or SGPT =\< 4 x ULN * Creatinine =\< 3 mg/dL or estimated creatinine clearance \>= 40ml/min Exclusion Criteria: * Uncontrolled bacterial, viral or fungal infection defined as currently taking medication and progression of clinical symptoms * Progressive hemato-lymphoid malignancy despite conventional therapy * Acute leukemia not in remission * Chronic myelogenous leukemia (CML) * Active central nervous system (CNS) involvement of the underlying malignancy * Human immunodeficiency virus (HIV) positive * Pregnant or lactating * Prior malignancy (EXCEPTION: diagnosed \> 5 years ago without evidence of disease, OR treated =\< 5 years ago but have a greater than 50% chance of life expectancy of \>= 5 years for that malignancy) * Have a psychiatric disorder(s) or psychosocial circumstance(s) which in the opinion of the primary physician would place the patient at an unacceptable risk from transplant * Ejection fraction \< 30%, or uncontrolled cardiac failure * Diffusing capacity of the lung for carbon monoxide (DLCO) \< 40% predicted * Total bilirubin \> 3 mg/dL * Serum glutamic oxaloacetic transaminase (SGOT) or serum glutamate pyruvate transaminase (SGPT) \> 4 x upper limit of normal (ULN) * Creatinine \> 2 mg/dL and an estimated creatinine clearance =\< 40 mL/min * Poorly controlled hypertension despite multiple antihypertensive medication OR * Karnofsky performance status (KPS) \< 60% * Note: Patients positive for hepatitis B and C will be evaluated on a case by case basis