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)) =\< …