• Patient must have a diagnosis of BRAF\^V600E/K-mutated metastatic melanoma

• Patient must have had documented radiographic or clinical evidence of progressive disease while on combination BRAF/MEK inhibitor therapy. For Phase 2 only, no more than one intervening therapy since progression on BRAF/MEK inhibitor therapy is allowed. Subjects who have evidence of progression while on, or within 4 weeks of completing, combination BRAF/MEK inhibitor therapy in the adjuvant setting will be eligible

• PHASE 2 ONLY: Patient must have EZH2 alteration (somatic mutation or copy number alteration). Can be performed on either archival or fresh specimen. EZH2 alterations need to be documented by a Clinical Laboratory Improvement Act (CLIA)/Clinical Laboratory Improvement Program (CLIP)-certified next generation sequencing platform (Foundation One, Tempus, Guardant360, etc.)

• PHASE 2 ONLY: Patient must have measurable disease

• PHASE 2 ONLY: Patient must have at least one tumor lesion amenable to biopsy. If possible, this lesion should be different from the lesion used for following tumor measurements but is not required

• PHASE 2 ONLY: Patient must agree to planned pre-treatment and planned on-treatment biopsy. A pre-treatment biopsy will be optional if patient has an archival tissue block or 5 formalin-fixed paraffin-embedded (FFPE) slides available from specimen used to document presence of eligible EZH2 alteration that is deemed adequate for evaluation

• Patient must be \>= 18 years

‣ Because no dosing or adverse event data are currently available on the use of tazemetostat in combination with dabrafenib and trametinib in patients \< 18 years of age, children are excluded from this study

• Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 50%)

• Patients with symptomatic central nervous system (CNS) metastases are eligible if previously treated with surgery and/or radiation with no evidence of radiologic CNS recurrence or progression for 4 weeks and on a stable/tapering dose of steroid for at least one week prior to start of study drug. Patients with new or progressive asymptomatic CNS metastases are eligible

• Hemoglobin \>= 9 g/dL

• Albumin \>= 2.5 g/dL

• Leukocytes \>= 3,000/mcL

• Absolute neutrophil count \>= 1,500/mcL

• Platelets \>= 100,000/mcL

• Serum bilirubin =\< 1.5 x institutional upper limit of normal (ULN) except subjects with known Gilbert's syndrome

• Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional ULN

• Creatinine =\< 1.5 mg/dL OR calculated creatinine clearance (Cockcroft-Gault formula) \>= 50 mL/min

• Glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2

• Patients with a prior (or concurrent, if enrolling in Phase 1) malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial

• Patient must be able to swallow and retain oral medication and must not have any clinically significant gastrointestinal abnormalities that may alter absorption such as malabsorption syndrome or major resection of the stomach or bowels

• Prothrombin time (PT)/international normalized ratio (INR) and partial thromboplastin time (PTT) =\< 1.3 x institutional ULN. Prophylactic low dose warfarin may be given to maintain central catheter patency

• The effects of tazemetostat, and the combination of tazemetostat, dabrafenib and trametinib on the developing human fetus are unknown. Women of childbearing potential and all male patients must agree to the following:

‣ For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of \< 1% per year during the treatment period, for 6 months after tazemetostat discontinuation, or for 6 months after discontinuation of the combination of tazemetostat, dabrafenib and trametinib. Should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately

• A woman is considered to be of childbearing potential if she is postmenarcheal, has not reached a postmenopausal state (\>= 12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus)

∙ Examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, established, proper use of hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices

∙ Due to the potential of enzyme induction with tazemetostat, female subjects who use hormonal contraceptives should use an additional barrier method of birth control while on study treatment and for 6 months after discontinuation of tazemetostat or the combination of tazemetostat, dabrafenib and trametinib

∙ The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical study and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post ovulation methods) and withdrawal are not acceptable methods of contraception

⁃ Women of childbearing potential must have a negative urine or serum pregnancy test at screening

⁃ For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures and agreement to refrain from donating sperm, as defined below:

• With female partners of childbearing potential or pregnant female partners, men must remain abstinent or use a condom during the treatment period and for at least 4 months after the last dose of study drug. Men must refrain from donating sperm during this same period. In addition, female partners of male subjects should adhere to the following:

‣ Intrauterine device (IUD) (must provide medical documentation of IUD)

⁃ Hormonal contraceptive (partner must be on a stable dose of the same hormonal contraceptive product for at least 4 weeks before receiving study drug) AND a condom (hormonal contraceptives must be supplemented with condoms)

∙ The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical study and the preferred and usual lifestyle of the patient. Periodic abstinence and withdrawal are not acceptable methods of contraception

• Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible

• Have progressed on, been intolerant to, is ineligible for, or has refused prior standard of care anti-PD-1 based immunotherapy