C-Met Inhibitor AMG 337, Oxaliplatin, Leucovorin Calcium, and Fluorouracil in Treating Patients With Advanced Stomach or Esophageal Cancer

Inclusion Criteria: * Patients must have a life expectancy \>= 12 weeks * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Women must not be pregnant or breast-feeding; all females of childbearing potential must have a blood test or urine study within 48 hours prior to randomization to rule out pregnancy; a female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) for the duration of study therapy and for 3 months after the last dose of AMG 337 plus mFOLFOX6 chemotherapy; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately; should a man impregnate or suspect that he has impregnated a woman while participating in this study, he should inform his treating physician immediately * Patients must NOT have a known immediate or delayed hypersensitivity reaction to drugs chemically related to fluorouracil, platins or their excipients nor have a known history of dihydropyrimidine dehydrogenase (DPD) deficiency * Patients must be able to swallow tablets whole * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Hemoglobin \>= 9 g/dL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 X institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 X institutional upper limit of normal (ULN) or =\< 5 X ULN if the patient has liver metastases * Creatinine =\< 1.5 X institutional ULN or creatinine clearance \>= 50 mL/min for patients with creatinine levels above institutional normal * Patients must NOT be taking current medications or substances that are inhibitors or inducers of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) * Patients with known human immunodeficiency virus (HIV) are not eligible if cluster of differentiation (CD)4 count is =\< 200 cell/mm\^3 or if receiving antiretroviral therapy * Patients must not have a pre-existing \> grade 1 (Common Terminology Criteria for Adverse Events version 4 \[CTCAEv4\]) motor or sensory neuropathy * Patients must not have an uncontrolled infection, active hepatic, biliary disease or other uncontrolled intercurrent illnesses including, but not limited to: uncontrolled ascites, uncontrolled hypertension, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia or psychiatric illness/addictive disorders that would limit compliance with study requirements * PHASE I: * Patients must have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria within 4 weeks prior to registration; patients must not have clinical or radiographic evidence of brain metastasis * Patients must have histologically or cytologically confirmed adenocarcinoma originating from anywhere in the gastrointestinal tract * Patients must have progression on standard therapies, for which effective therapy is not available (or patients are not a candidate for or are intolerant of such therapies) * Patients must NOT have previous or concurrent malignancy; exceptions are made for patients who meet any of the following conditions: * Non-melanoma skin cancer, in situ cervical cancer, breast cancer in situ, or superficial bladder cancer (noninvasive papillary carcinoma or carcinoma in situ) * Prior malignancy completely excised or removed and patient has been continuously disease free for \> 5 years from registration * Patients may have received prior surgery or radiotherapy if \> 4 weeks prior to registration and patient does not have any unresolved or unstable serious toxicity * Patients must not be receiving an investigational agent concurrently and must not have received any other investigational agents within 4 weeks prior to randomization * PHASE II: * Patients must have measurable disease by RECIST 1.1 criteria within 4 weeks prior to registration to Step 0; patients must not have clinical or radiographic evidence of brain metastasis because of their poor prognosis * Patient disease status must be as follows: * Adenocarcinoma or poorly differentiated carcinoma of the stomach, esophagus or gastroesophageal (GE) junction * Histologically or cytologically confirmed Her2/neu negative cancer prior to pre-registration; Her2/neu status must be determined by a Clinical Laboratory Improvement Amendments (CLIA) certified laboratory * Locally advanced or metastatic disease that is inoperable and not amenable to curative therapy; linitis plastica is permitted * Patient must NOT have gastric carcinoid, sarcomas or squamous cell carcinoma * Patients may have received prior surgery or radiotherapy if \> 4 weeks prior to registration to Step 0 and patient does not have any unresolved or unstable serious toxicity * Patients must NOT have previous or concurrent malignancy; exceptions are made for patients who meet any of the following conditions: * Non-melanoma skin cancer, in situ cervical cancer, breast cancer in situ, or superficial bladder cancer (noninvasive papillary carcinoma or carcinoma in situ) * Prior malignancy completely excised or removed and patient has been continuously disease free for \> 5 years from registration to Step 0 * Patients must not be receiving an investigational agent concurrently and must not have received any other investigational agents within 4 weeks prior to registration to Step 0 * Patient may not have received prior chemotherapy for advanced disease or prior oxaliplatin or anti-MET therapy * Previous neo-adjuvant or adjuvant treatment is allowed provided that it was discontinued \>= 6 months prior to registration to Step 0 * Patients must have paraffin-embedded tumor specimen available for submission for central determination of MET expression status * NOTE: it is recommended that patients not be pre-registered until the required tumor specimens are on hand and ready for submission; if submission of tissue will be submitted more than 5 working days after pre-registration, immediately notify the receiving laboratory * Patients must have had MET expression status determined by the Immunohistochemistry Laboratory and Image Analysis Laboratory in the Department of Pathology of The University of Texas MD Anderson Cancer Center in Houston: the report from the central laboratory indicates tumor tissue has MET * NOTE: for this protocol, 'MET High' will be defined as: \>= 50% tumor cells with a staining intensity of 2+ or 3+ in IHC utilizing the CONFIRM anti-total MET, SP44, rabbit monoclonal primary antibody