Brigatinib and Binimetinib in Treating Patients With Stage IIIB-IV ALK or ROS1-Rearranged Non-small Cell Lung Cancer

Inclusion Criteria: * Participants must have histologically or cytological confirmed stage IIIB/IV NSCLC * Documented ALK-rearrangement (or ROS1- rearrangement) break-apart fluorescence in situ hybridization (FISH) (in \>= 15% or tumor cells), or next generation sequencing assay performed on tumor sample or cell free DNA in a Clinical Laboratory Improvement Act (CLIA)-approved laboratory * At least one prior ALK or ROS1 targeted tyrosine kinase inhibitor (TKI). With progression or intolerance of most recent regimen * Dose Escalation Phase Only: At least one prior ALK or ROS1 targeted TKI. With progression or intolerance of most recent regimen. * Dose Expansion Phase Only: In addition to the criteria in Exclusion Criteria # 3, ALK+ patients with no prior chemotherapy, immunotherapy, radiation therapy or other systemic therapy are also allowed. * Measurable or evaluable disease defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria * Age \>= 18 years * Life expectancy of at least 3 months * Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 * Leukocytes \>= 3,000/microliter (mcL) * Absolute neutrophil count \>=1,500/mcL * Platelets \>= 75,000/mcL * Hemoglobin (Hgb) \>= 9 gm/dL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase (SGOT)) =\< 5 x institutional upper limit of normal * Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase (SGPT)) =\< 5 x institutional upper limit of normal if there are liver metastases * Creatinine within normal institutional limits OR * Creatinine clearance \>= 50 mL/min/1.73 m\^2 for participants with creatinine levels above institutional normal * Female participants who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent through 4 months after the last dose of the study drugs, or * Agree to completely abstain from heterosexual intercourse * Male participants, even if surgically sterilized (i.e., status post-vasectomy), who: * Agree to practice effective barrier contraception during the entire study treatment period and through 4 months after the last dose of the study drugs, or * Agree to completely abstain from heterosexual intercourse * Negative pregnancy testing required at screening and cycle 1 day 1 for women of childbearing potential * Ability to understand a written informed consent document, and willingness to sign it * Able to swallow and retain orally administered medication and does not have any clinically significant gastrointestinal abnormalities that may alter absorption such as malabsorption syndrome or major resection of the stomach or bowels * Participant is deemed by the investigator to have the initiative and means to be compliant with the protocol (treatment and follow-up) Exclusion Criteria: * Is currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 7 days for prior TKI and 14 days for chemotherapy or radiation and 30 days for prior immunotherapy before the first dose of treatment * Prior treatment with either study drug as most recent treatment or prior discontinuation of either study drug due to toxicity * Known hypersensitivity to any study drug components * Known history of interstitial lung disease or interstitial pneumonitis, including clinically significant radiation pneumonitis * Known history of myositis * History of pancreatitis * History or current evidence of retinal vein occlusion (RVO) or current risk factors to RVO (e.g. uncontrolled glaucoma or ocular hypertension, history of hyperviscosity or hypercoagulability syndromes); history of retinal degenerative disease * Impaired cardiovascular function or clinically significant cardiovascular diseases, including any of the following: * Symptomatic chronic heart failure grade \>= 2, history or current evidence of clinically significant cardiac arrhythmia and/or conduction abnormality \< 6 months prior to screening except atrial fibrillation and paroxysmal supraventricular tachycardia. * Have significant, uncontrolled, or active cardiovascular disease, specifically including, but not restricted to: * Myocardial infarction (MI) within 6 months prior to the first dose of brigatinib * Unstable angina within 6 months prior to first dose * Symptomatic congestive heart failure requiring treatment (New York Heart Association grade \>= 2), history or current evidence of clinically significant cardiac arrhythmia and/or conduction abnormality =\< 6 months prior to Screening except atrial fibrillation and paroxysmal supraventricular tachycardia * Congestive heart failure requiring treatment (New York Heart Association grade \>= 2) * Left ventricular ejection fraction (LVEF) \< 50% as determined by multigated acquisition scan (MUGA) or echocardiogram (ECHO) * QT interval corrected for heart rate using the Fridericia formula (QTcF) \> 480 msec; * History of clinically significant atrial arrhythmia * Any history of ventricular arrhythmia * Cerebrovascular accident or transient ischemic attack within 6 months prior to first dose * Uncontrolled hypertension defined as persistent elevation of systolic blood pressure \>= 150 mmHg or diastolic blood pressure \>= 100 mmHg despite current therapy * History of chronic inflammatory bowel disease or Crohn?s disease requiring medical intervention (immunomodulatory or immunosuppressive medications or surgery) =\< 12 months prior to first dose of study treatment * Impaired gastrointestinal (GI) function or disease that may significantly alter the absorption of binimetinib or brigatinib (e.g., ulcerative diseases, uncontrolled vomiting, malabsorption syndrome, small bowel resection with decreased intestinal absorption) * History of thromboembolic or cerebrovascular events =\< 6 months prior to starting study treatment, including transient ischemic attacks, cerebrovascular accidents, deep vein thrombosis or pulmonary emboli * Current neuromuscular disorder(s) associated with elevated creatine kinase (CK) (e.g., inflammatory myopathies, muscular dystrophy, amyotrophic lateral sclerosis, spinal muscular atrophy) * Major surgery =\< 3 weeks prior to starting study drugs, or persistent/ongoing side effects of major surgery * Pregnant or nursing (lactating) women * Other severe, acute, or chronic medical conditions including uncontrolled diabetes mellitus or psychiatric conditions or laboratory abnormalities that, in the opinion of the investigator, may increase the risk associated with study participation or may interfere with ability to participate actively in the study * History of another malignancy. Participants with fully resected non-melanoma skin cancer, indolent early stage second malignancies, or who have been disease free for \> three years may be eligible following discussion with study principal investigator (PI) * Any symptomatic brain metastasis * Note: Participants previously treated or untreated for this condition who are asymptomatic in the absence of corticosteroid and anti-epileptic therapy are allowed. Brain metastases must be stable for \>= 2 weeks, with imaging (e.g., magnetic resonance imaging \[MRI\] or computed tomography \[CT\]) demonstrating no current evidence of progressive brain metastases at screening) * Participants receiving any medications or substances that are inhibitors or inducers of CYP3A enzyme(s) and CYP2C8 within 14 days of enrollment are ineligible * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions. In addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy