T-Cell Replete Haploidentical Donor Hematopoietic Stem Cell Plus Natural Killer (NK) Cell Transplantation in Patients With Hematologic Malignancies Relapsed or Refractory Despite Previous Allogeneic Transplant

Inclusion Criteria - for transplant recipient:

* Age less than 21 years.
* One of the following hematologic malignancies that has relapsed or remains refractory after prior allogeneic HCT:

  * Acute lymphoblastic leukemia (ALL)
  * Acute myeloid leukemia (AML) (including myeloid sarcoma)
  * Chronic myelogenous leukemia (CML), juvenile myelomonocytic leukemia (JMML), myelodysplastic syndrome (MDS), Hodgkin or non-Hodgkin lymphoma (NHL)
* Has a suitable single haplotype matched (≥ 3 of 6) family member donor.
* Does not have any other active malignancy other than the one for which this transplant is indicated.
* If prior central nervous system (CNS) leukemia, it must be treated and have no evidence of CNS disease
* Does not have current uncontrolled bacterial, fungal, or viral infection per the judgment of the principal investigator.
* Patient must fulfill pre-transplant evaluation:

  * Left ventricular ejection fraction greater than 40%, or shortening fraction greater than or equal to 25%.
  * Creatinine clearance or Glomerular Filtration Rate of ≥70 ml/min/1.73m\^2.
  * Forced vital capacity (FVC) ≥ 40% of predicted value; or pulse oximetry ≥ 92% on room air if patient is unable to perform pulmonary function testing.
  * Karnofsky or Lansky (age-dependent) performance score ≥ 50.
  * Total bilirubin ≤ 1.5 times the upper limit of normal for age.
  * Alanine aminotransferase (ALT) ≤ 3 times the upper limit of normal for age.
  * Not pregnant. If female with child bearing potential, must be confirmed by negative serum or urine pregnancy test within 14 days prior to enrollment.
  * Not breast feeding.
  * Does not have active acute bronchiolitis obliterans or bronchiolitis obliterans organizing pneumonia.

Inclusion Criteria - for donor:

* At least single haplotype matched (≥ 3 of 6) family member,
* At least 18 years of age.
* Human immunodeficiency virus (HIV) negative.
* Not pregnant as confirmed by negative serum or urine pregnancy test within 14 days prior to enrollment (if female).
* Not breast feeding.
* A suitable donor is identified as either:

  * Has completed the process of donor eligibility determination as outlined in 21 CFR 1271 and agency guidance; OR
  * Does not meet 21 CFR 1271 eligibility requirements, but has a declaration of urgent medical need completed by the principal investigator or physician sub-investigator per 21 CFR 1271.

Exclusion Criteria:

* Does not meet above inclusion criteria.