Immune cell therapy tested for tough childhood cancers

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

Inclusion Criteria: * SCREENING: Patients with relapsed or refractory solid tumors and without known curative therapy or therapy proven to proven to prolong survival with acceptable quality of life. * SCREENING: Patients older than 21 years must have a solid tumor considered by study doctor to be of the childhood cancer type. * SCREENING: Performance level as measured by Karnofsky \>= 60% for patients \> 16 years of age or Lansky \>= 60% for patients =\< 16 years of age. * SCREENING: Documentation of measurable or evaluable non-measurable disease. * SCREENING: At least one documented histological verification of solid tumor diagnosis. Can be from original diagnosis or more recent. * ENROLLMENT: Patient must have fully recovered (i.e. returned to baseline) from the clinically significant acute treatment-related toxicities of all prior treatments prior to beginning treatment on this protocol with exceptions of cytopenias resulting from persistent disease, hearing loss and alopecia. * ENROLLMENT: Performance level as measured by Karnofsky \>= 60% for patients \> 16 years of age or Lansky \>= 60% for patients =\< 16 years of age. * ENROLLMENT: Creatinine clearance \>= 60 mL/min/1.73m\^2 (calculated by 24 hour \[h\] urine collection or nuclear glomerular filtration rate \[GFR\] scan if 24 h collection is not possible) or a serum creatinine based on age and gender as follows: * Age, maximum serum creatinine (mg/dL): * 1 month to \< 6 months, male 0.4, female 0.4; * 6 months to \< 1 year, male 0.5, female 0.5; * 1 to \< 2 years, male 0.6, female 0.6; * 2 to \< 6 years, male 0.8, female 0.8; * 6 to \< 10 years, male 1, female 1; * 10 to \< 13 years, male 1.2, female 1.2; * 13 to \< 16 years, male 1.5, female 1.4; * \>= 16 years, male 1.7, female 1.4. * ENROLLMENT: Adequate liver function, defined as: total bilirubin =\< 2 mg/dl * ENROLLMENT: Adequate liver function, as defined as serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 2.5 x upper limit of normal (ULN) for age (unless Gilbert's disease or abnormal liver function due to primary disease). * ENROLLMENT: Evidence of adequate bone marrow function (defined by absolute neutrophil count \>= 750), unless patient has documented tumor metastasis to the bone marrow or other condition that results in cytopenia without abnormal marrow function. * ENROLLMENT: Evidence of adequate bone marrow function (defined by platelets \>= 50,000), unless patient has documented tumor metastasis to the bone marrow or other condition that results in cytopenia without abnormal marrow function. * ENROLLMENT: Pulmonary symptoms controlled by medication and pulse oximetry \>= 92% on room air. * ENROLLMENT: Sexually active males and females of childbearing potential must agree to use a form of contraception considered effective and medically acceptable by the investigator. (Non-childbearing potential defined as pre-menarche, greater than one year post-menopausal or surgically sterilized). * ENROLLMENT: Confirmation that a cord blood donor which is matched with the recipient at a 4, 5, or 6/6 human leukocyte antigen (HLA) class I (serological) and HLA class II (molecular) antigens. * ENROLLMENT: Signed informed consent and if applicable pediatric assent. Exclusion Criteria: * SCREENING: Primary tumors of the central nervous system. * SCREENING: Chronic corticosteroid dependence that is unable to be weaned to discontinue. * SCREENING: Determined by study doctor that patient is unlikely to meet inclusion criteria after screening. * ENROLLMENT: Uncontrolled arrhythmias or uncontrolled symptoms of cardiac disease noted by screening history and physical. Patients with known cardiac dysfunction should have an ejection fraction (EF) \> 40% documented by echocardiogram (ECHO). * ENROLLMENT: Patients where the burden of pulmonary metastasis, location, or bulkiness of disease may cause high morbidity if localized swelling such as causing uncontrolled symptoms, oxygen dependence, or location near a major bronchi as determined by investigator. * ENROLLMENT: Pregnant females. * ENROLLMENT: Any uncontrolled systemic infection.