Genetically Modified T-Cell Therapy in Treating Patients With Advanced ROR1+ Malignancies

Inclusion Criteria: INCLUSION CRITERIA FOR PATIENTS WITH CLL, MCL OR ALL (COHORT A) * CLL who are beyond first remission and who have failed combination chemoimmunotherapy with regimens containing a purine analogue and anti-CD20 antibody, or who have failed tyrosine kinase or phosphatidylinositol 3 (PI3) kinase inhibitors, or who were not eligible for or declined such therapy; patients with fludarabine refractory disease are eligible * Mantle cell lymphoma patients who are beyond first remission and previously treated with chemoimmunotherapy; patients who have relapsed following autologous hematopoietic cell transplant (HCT) are eligible * ALL patients who have relapsed or have residual disease following treatment with curative intent; ALL patients must have ROR1 expressed on \> 90% of the leukemia blasts to be eligible * Confirmation of diagnosis by internal pathology review of initial or subsequent biopsy or other pathologic material at the Fred Hutchinson Cancer Research Center (FHCRC)/Seattle Cancer Care Alliance (SCCA) * Evidence of ROR1 expression by immunohistochemistry or flow cytometry on any prior or current tumor specimen * Karnofsky performance status \>= 70% * Negative pregnancy test for women of childbearing potential; subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year * Fertile male and female patients must be willing to use a contraceptive method before, during, and for at least two months after the T cell infusion * Ability to understand and provide informed consent INCLUSION CRITERIA FOR PATIENTS WITH NSCLC OR TNBC (COHORT B): * Patients with non-small cell lung cancer that is metastatic or inoperable and who have been treated with at least one line of prior therapy or declined conventional therapy * Patients with known epidermal growth factor receptor (EGFR) or anaplastic lymphoma kinase (ALK) mutations must have been treated on at least one line of molecularly targeted therapy (e.g., erlotinib, crizotinib) * Patients must have measurable disease by at least one of the criteria below: * Extra skeletal disease that can be accurately measured in at least one dimension as \>= 10 mm with conventional computed tomography (CT) techniques as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Skeletal or bone-only disease measurable by fludeoxyglucose F 18 (FDG) positron emission tomography (PET) imaging * ROR1 expression in \> 20% of the primary tumor or metastasis by immunohistochemistry (IHC) * Karnofsky performance status of \>= 70% * Patients must be off chemotherapy for a minimum of 3 weeks prior to start of treatment; targeted therapies must be stopped at least 3 days prior to start of lymphodepletion * Negative pregnancy test for women of childbearing potential; subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year * Fertile male and female patients must be willing to use a contraceptive method before, during and for at least two months after the T cell infusion * Ability to understand and provide informed consent INCLUSION CRITERIA FOR TNBC: * Histologically confirmed diagnosis of metastatic TNBC; i.e. breast cancer that is estrogen receptor (ER) negative (=\< 10%), progesterone receptor (PR) negative (=\< 10%), and human epidermal growth factor receptor 2 (HER2) negative (0 or 1+ by immunohistochemistry or negative for gene amplification by fluorescence in situ hybridization \[FISH\]) * Patients must have measurable disease by at least one of the criteria below: * Extra skeletal disease that can be accurately measured in at least one dimension as \>= 10 mm with conventional CT techniques as defined by RECIST 1.1 * Skeletal or bone-only disease measurable by FDG PET imaging * Patients must have received standard adjuvant, neoadjuvant, and/or metastatic chemotherapy per National Comprehensive Cancer Network (NCCN) or institutional practice; no maximum on number of prior systemic treatment regimens * Patients may receive agents to protect against skeletal related complications such as zoledronic acid or denosumab * ROR1 expression in \> 20% of the primary tumor or metastasis by IHC * Karnofsky performance status of \>= 70% * Patients must be off chemotherapy for a minimum of 3 weeks prior to planned leukapheresis * Negative pregnancy test for women of childbearing potential; subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year * Fertile male and female patients must be willing to use a contraceptive method before, during and for at least two months after the T cell infusion * Ability to understand and provide informed consent Exclusion Criteria: EXCLUSION CRITERIA FOR PATIENTS WITH CLL, MCL, OR ALL (COHORT A) * Treatment with other investigational agent(s) within 30 days of planned lymphodepletion * Patients requiring ongoing daily corticosteroid therapy at a dose of \> 15 mg of prednisone per day (or equivalent); pulsed corticosteroid use for disease control is acceptable * Active autoimmune disease requiring immunosuppressive therapy * Serum creatinine \> 2.5 mg/dL * Serum glutamic oxaloacetic transaminase (SGOT) \> 5 x upper limit of normal * Bilirubin \> 3.0 mg/dL * Patients with clinically significant pulmonary dysfunction, as determined by medical history and physical exam should undergo pulmonary function testing; those with a forced expiratory volume in 1 second (FEV1) of =\< 65% or diffusion capacity of the lung for carbon monoxide (DLCO) (corrected) \< 40% will be excluded * Significant cardiovascular abnormalities as defined by any one of the following: congestive heart failure, clinically significant hypotension, symptomatic coronary artery disease, or a documented ejection fraction of \< 45%; any patient with an ejection fraction (EF) of 45-49% must receive clearance by a cardiologist to be eligible for the trial * Patients who are human immunodeficiency virus (HIV) seropositive * Uncontrolled active infection (bacterial, viral, fungal, mycobacterial) not responding to treatment with intravenous antibiotics, antiviral or antifungal agents, or long-term treatment with oral agents * Women who are breast-feeding * Patients who have contraindication to cyclophosphamide chemotherapy * Known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone potentially curative therapy * Untreated central nervous system (CNS) metastases and/or carcinomatous meningitis; subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to enrollment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases EXCLUSION CRITERIA FOR PATIENTS WITH NSCLC OR TNBC (COHORT B): * Absolute neutrophil count (ANC) \< 1000/mm\^3 * Hemoglobin (Hgb) \< 9 mg/dl (transfusion permitted to achieve this) * Platelet count \< 75,000/mm\^3 * Treatment with other investigational agent(s) within 30 days of planned lymphodepletion * Patients requiring ongoing daily corticosteroid therapy at a dose of \> 15 mg of prednisone per day (or equivalent); pulsed corticosteroid use for disease control is acceptable * Active autoimmune disease requiring immunosuppressive therapy * Serum creatinine \> 2.5 mg/dL * SGOT \> 5 x upper limit of normal * Bilirubin \> 3.0 mg/dL * Patients with clinically significant pulmonary dysfunction, as determined by medical history and physical exam should undergo pulmonary function testing; those with an FEV1 of =\< 65% or DLCO (corrected) \< 40% will be excluded * Significant cardiovascular abnormalities as defined by any one of the following: congestive heart failure, clinically significant hypotension, symptomatic coronary artery disease, or a documented ejection fraction of \< 45%; any patient with an EF of 45-49% must receive clearance by a cardiologist to be eligible for the trial * Patients who are HIV seropositive * Uncontrolled active infection (bacterial, viral, fungal, mycobacterial) not responding to treatment with intravenous antibiotics, antiviral or antifungal agents, or long-term treatment with oral agents * Women who are breastfeeding * Patients who have contraindication to cyclophosphamide chemotherapy * Known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone potentially curative therapy * Untreated central nervous system (CNS) metastases and/or carcinomatous meningitis; subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to enrollment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases EXCLUSION CRITERIA FOR TNBC: * ANC \< 1000/mm\^3 * Hgb \< 9 mg/dl (transfusion permitted to achieve this) * Platelet count \< 75,000/mm\^3 * Treatment with other investigational agent(s) within 30 days of planned lymphodepletion * Patients requiring ongoing daily corticosteroid therapy at a dose of \> 15 mg of prednisone per day (or equivalent); pulsed corticosteroid use for disease control is acceptable * Active autoimmune disease requiring immunosuppressive therapy * Serum creatinine \> 2.5 mg/dL * SGOT \> 5 x upper limit of normal * Bilirubin \> 3.0 mg/dL * Patients with clinically significant pulmonary dysfunction, as determined by medical history and physical exam should undergo pulmonary function testing; those with an FEV1 of =\< 65% or DLCO (corrected) \< 40% will be excluded * Significant cardiovascular abnormalities as defined by any one of the following: congestive heart failure, clinically significant hypotension, symptomatic coronary artery disease or a documented ejection fraction of \< 45%; any patient with an EF of 45-49% must receive clearance by a cardiologist to be eligible for the trial * Patients who are HIV seropositive * Uncontrolled active infection (bacterial, viral, fungal, mycobacterial) not responding to treatment with intravenous antibiotics, antiviral or antifungal agents, or long-term treatment with oral agents * Breast-feeding women * Patients who have contraindication to cyclophosphamide chemotherapy * Has a known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone potentially curative therapy * Untreated central nervous system (CNS) metastases and/or carcinomatous meningitis; subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to enrollment and any neurologic symptoms have returned to baseline) and have no evidence of new or enlarging brain metastases