Study of Chemotherapy, With or Without Binimetinib in Advanced Biliary Tract Cancers in 2nd Line Setting (A ComboMATCH Treatment Trial)

Inclusion Criteria: * Patient must have enrolled onto EAY191 and must have been given a treatment assignment to ComboMATCH to EAY191-A6 based on the presence of an actionable mutation as defined in EAY191 * GENERAL COMBOMATCH EAY191: * Patients must be registered to the ComboMATCH Registration Protocol (EAY191) * Patients must have RAS/RAF/MEK/ERK mutations as determined by the ComboMATCH screening assessment * Patients must not have BRAF V600E as determined by the ComboMATCH screening assessment * Patients must have disease that can be safely biopsied and agree to a pre-treatment biopsy or have archival tissue available from within 12 months prior to registration on the ComboMATCH Registration Trial (EAY191). * Please note the current actionable marker of interest (aMOI)/actionable alteration list for this treatment trial can be found on the Cancer Trials Support Unit (CTSU) website * Please note novel/Dynamic aMOI can be submitted for review per the process described in the ComboMATCH Registration Protocol * EAY191-A6 REGISTRATION: * Participants must have histologically confirmed BTC (intrahepatic cholangiocarcinoma \[IHC\], extrahepatic cholangiocarcinoma \[EHC\] or gallbladder cancer \[GBC\]) that is unresectable or recurrent with a confirmed RAS/RAF/MEK/ERK pathway mutation via any Clinical Laboratory Improvement Act (CLIA)-certified method. BRAFV600E mutations are not eligible due to other ongoing/upcoming studies in this disease cohort * Tumor tissue must be available: * Adequate archival tumor specimen (obtained within 12 months of EAY191 registration which has not had a Response Evaluation Criteria in Solid Tumors (RECIST) response, complete response (CR) or partial response (PR), to any intervening therapy after collection of the tissue) must be available with formalin-fixed paraffin-embedded tumor tissue (blocks or slides) OR * Consent to a new tumor tissue biopsy which is not a representative target lesion. This lesion must be amenable to a minimal risk image-guided or direct vision biopsy * A new biopsy is preferred but is not required for enrollment in EAY191-A6 if sufficient archival tissue is available as described above. * Measurable disease per RECIST 1.1 Of note, in the case when a baseline biopsy is done after scans are obtained, a lesion separate from one that is biopsied needs to be measurable per RECIST 1.1. All radiologic studies must be performed within 28 days prior to registration * Progression of disease on gemcitabine based first-line regimen (i.e. only one prior line of therapy is permitted) * No systemic anti-cancer therapy within 4 weeks of registration to EAY191-A6 * No prior MEK inhibitor therapy * No prior history of treatment with a direct and specific inhibitor of KRAS * Patients who only received radio-sensitizing chemotherapy with fluorouracil (5-FU) or capecitabine are eligible, but need to have received and failed first-line systemic chemotherapy upon recurrence. Peri-operative systemic 5-FU/capecitabine and/or oxaliplatin, is allowed if it's been more than 12 months of registration to EAY191-A6 * No major surgery within 4 weeks (excluding placement of vascular access) of registration to EAY191-A6 * No minor surgery within 2 weeks of registration to EAY191-A6 * No palliative radiotherapy within 1 week of registration to EAY191-A6 * Not pregnant and not nursing, because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown * Therefore, for women of childbearing potential only, a negative pregnancy test done =\< 14 days prior to registration is required * Adequate contraception is needed for at least 30 days after the last dose of binimetinib and breastfeeding should be discontinued for at least 3 days after the last dose of binimetinib. For FOLFOX regimen, 9 months is recommended for contraception after last dose of oxaliplatin for females of childbearing potential and 6 months for males * Age \>= 18 years * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Absolute neutrophil count (ANC) \>= 1,000/mm\^3, no growth factor within 14 days of 1st dose * Platelet count \>= 75,000/mm\^3 * Creatinine \< 1.6 x upper limit of normal (ULN) OR * Calculated (Calc.) creatinine clearance \>= 50 mL/min, as calculated by the Cockcroft-Gault formula * Total bilirubin =\< 2.0 x upper limit of normal (ULN); Patients with Gilbert syndrome may enroll if \< 3.0 x ULN * Aspartate aminotransferase (AST) /alanine aminotransferase (ALT) =\< 5.0 x upper limit of normal (ULN) * Hemoglobin \>= 8 g/dL, no transfusion within 7 days of 1st dose * Creatine phosphokinase =\< 2.5 x ULN * High blood pressure more than 160/90 despite treatment are ineligible * No history of interstitial lung disease. No history of idiopathic pulmonary fibrosis, organizing pneumonia (e.g., bronchiolitis obliterans), drug-induced pneumonitis, or idiopathic pneumonitis, or evidence of active pneumonitis on screening chest computed tomography (CT) scan * Patients should not have history of bowel perforation or intestinal fistulas in the last 6 months * Patients with the inability to swallow oral medications or impaired gastrointestinal absorption due to gastrectomy or active inflammatory bowel disease are ineligible * Must have adequate cardiac function with left ventricular ejection fraction \>= 50% by echocardiography (ECHO) or multiple-gated acquisition (MUGA) scan. Patients with congenital long QT syndrome are not permitted * No history of prolonged QTc or at risk for prolonged QTc due to any reason (for example, concomitant medications during or before chemotherapy that may increase the risk of prolonged QTc), uncontrolled congestive heart failure, prior history of class III/ IV cardiac disease (New York Heart Association \[NYHA\]), myocardial infarction within the last 6 months, unstable arrhythmias, unstable angina or severe obstructive pulmonary disease * No active skin disorder that has required systemic therapy within the past 1 year * No history of rhabdomyolysis * No concurrent ocular disorders, including: * Patients with history of glaucoma, history of retinal vein occlusion (RVO), predisposing factors for RVO, including but not limited to uncontrolled hypertension, uncontrolled diabetes * Patients with history of retinal pathology or evidence of visible retinal pathology that is considered a risk factor for RVO, intraocular pressure \> 21 mm Hg as measured by tonometry, or other significant ocular pathology, such as anatomical abnormalities that increase the risk for RVO * Patients with a history of corneal erosion (instability of corneal epithelium), corneal degeneration, active or recurrent keratitis, and other forms of serious ocular surface inflammatory conditions * Patients with known or at risk for retinopathies, uveitis or retinal vein occlusion * No patients with a history of hypersensitivity to any of the inactive ingredients in binimetinib, nor known severe allergic reactions or hypersensitivity of 5-FU, leucovorin (LV) or oxaliplatin will be allowed to participate in this study for safety concerns * No other medical condition (e.g. cardiac, gastrointestinal, pulmonary, psychiatric, neurological, genetic, etc.) that in the opinion of the investigator would places the subject at unacceptably high risk for toxicity * No prior allogeneic stem cell or solid organ transplantation * Central nervous system (CNS) metastases must have been treated with local therapy (surgery, radiation, ablation) with systemic steroids tapered to a physiologic dose (10 mg or prednisone equivalent or less) * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients must not have grade 2 neuropathy or greater, within 14 days prior to registration