Nivolumab With Trametinib and Dabrafenib, or Encorafenib and Binimetinib in Treating Patients With BRAF Mutated Metastatic or Unresectable Stage III-IV Melanoma

Inclusion Criteria:

* Histologically confirmed metastatic melanoma (stage IV) or unresectable Stage III that have progressed on or after receiving prior PD-1 directed therapy; only patients with BRAF V600 mutated melanoma are eligible; please note that patients with brain metastasis are not required to have prior PD-1
* Prior therapies for metastatic melanoma are allowed, including chemo-, cytokine-, immuno, biological and vaccine-therapy as long as they did not include BRAFi, MEKi; patients who have progressed on or after receiving anti-PD-1therapy in the adjuvant setting are also allowed; prior ipilimumab and/or PD-1 directed therapy will be allowed with a washout period of 2 weeks and if all autoimmune adverse events have resolved to grade 1 (except endocrine abnormalities that require continuous replacement)
* Evidence of evaluable disease
* Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* Patients with melanoma brain metastases are allowed regardless of prior PD-1 exposure. Subjects with brain metastases are eligible if:

  * Metastases have been treated and there is no magnetic resonance imaging (MRI) evidence of progression for 2 weeks after treatment is complete and within 14 days of the first dose of nivolumab administration; or
  * If they are untreated but asymptomatic; or
  * If they are untreated and symptomatic but symptoms are controlled on stable or decreasing doses of steroids for 14 days prior to drug administration; or
  * If they have untreated leptomeningeal disease (LMD) as long as they fulfill all other eligibility requirements.
  * Note: Patients are excluded if they require high doses of systemic corticosteroids (\> 8 mg equivalent of dexamethasone) to control central nervous system (CNS) symptoms.
* White blood cells (WBC) \>= 2000 /uL (within one week prior to registration)
* Neutrophils \>= 1500 /uL (within one week prior to registration)
* Platelets \>= 100 x 10\^3 /uL (within one week prior to registration)
* Hemoglobin \> 9.0 g/dL (within one week prior to registration)
* Serum creatinine =\< 1.5 x upper limit of normal (ULN) or creatinine clearance (CrCl) \>= 40 mL/min (if using the Cockcroft-Gault formula) (within one week prior to registration)
* Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 3 x ULN (within one week prior to registration)
* Total bilirubin =\< 1.5 x ULN (except subjects with Gilbert syndrome, who can have total bilirubin \< 3.0 mg/dL) (within one week prior to registration)
* Women of childbearing potential (WOCBP) must be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 23 weeks (30 days plus the time required for nivolumab to undergo five half-lives) after the last dose of investigational drug; WOCBP are those who have not been surgically sterilized or have not been free from menses for \> 1 year
* Women of childbearing potential must have a negative serum or urine pregnancy test within 72 hours prior to the start of nivolumab
* Men who are sexually active with WOCBP must use any contraceptive method with a failure rate of less than 1% per year; men receiving nivolumab and who are sexually active with WOCBP will be instructed to adhere to contraception for a period of 31 weeks after the last dose of investigational product; women who are not of childbearing potential (i.e., who are postmenopausal or surgically sterile) and azoospermic men do not require contraception
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Patients that had grade 3/4 immune-related adverse events (AEs) on ipilimumab or nivolumab that required more than 12 weeks of immune suppression with corticosteroids
* History of interstitial lung disease or pneumonitis
* History of known glucose-6-phosphate dehydrogenase (G6PD) deficiency
* Active, known or suspected autoimmune disease; subjects are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism and/or hypophysitis due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger
* Require systemic treatment with either corticosteroids (\> 8 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration; inhaled or topical steroids and adrenal replacement doses are permitted in the absence of active autoimmune disease
* Known history of a positive test for hepatitis B virus or hepatitis C virus indicating acute or chronic infection
* Known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS); HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with dabrafenib and trametinib; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated
* History of allergy or adverse drug reaction to the study drug components (nivolumab, dabrafenib, or trametinib) or drugs of similar chemical or biologic composition; patients with a history of severe hypersensitivity reaction to any monoclonal antibody should also be excluded
* Patients who have had chemotherapy or radiotherapy within 2 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier
* Uncontrolled intercurrent illness (requiring IV antibiotic treatment) including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant and/or breastfeeding women are excluded from this study; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with nivolumab, dabrafenib, and trametinib, breastfeeding should be discontinued if the mother is treated with nivolumab, dabrafenib, and trametinib; these potential risks may also apply to other agents used in this study