Testing the Addition of Tazemetostat to the Immunotherapy Drug, Pembrolizumab (MK-3475), in Advanced Urothelial Carcinoma

Inclusion Criteria:

* Patients must have pathologically confirmed urothelial carcinoma

  * Note: Patients with mixed histology (with predominant urothelial carcinoma) are eligible
* Patients must have locally advanced or metastatic disease with either:

  * Arm A: Disease progression during or following (within 12 months) platinum-based chemotherapy (cisplatin or carboplatin)

    * Note: No minimum number of cycles on platinum-based chemotherapy are required. Patients who have had multiple rounds of platinum-based chemotherapy with events of intermittent progressive disease (PD) are eligible as long as progression has been confirmed while on or within 12 months from platinum based therapy OR
  * Arm B: Platinum ineligible status (i.e., patients unable to receive platinum-containing chemotherapy, in the opinion of the treating investigator, regardless of PD-L1 status) or chemotherapy ineligible status (i.e., patients unable to receive treatment with any chemotherapy, in the opinion of the treating investigator, regardless of PD-L1 status)

    * Note: The eligible patient population for Arm B was revised in protocol amendment 10 from 'cisplatin ineligible patients with PD-L1 positive disease' to 'platinum ineligible patients (i.e., unable to receive platinum-containing chemotherapy, in the opinion of the treating investigator, regardless of PD-L1 status) or chemotherapy ineligible patients (i.e., unable to receive treatment with any chemotherapy, in the opinion of the treating investigator, regardless of PD-L1 status).' This revision was made to align with the updated United States (U.S.) Food and Drug Administration (FDA) indications for pembrolizumab (MK-3475) that were released in August of 2021 for patients with locally advanced or metastatic urothelial carcinoma
* All patients must have measurable disease in accordance with RECIST criteria version (v) 1.1

  * Note: radiological evaluation should occur within 28 days prior to study registration
* Patients must be naive to prior PD-L1 or EZH2 inhibitors

  * Note: patients in Arm A should have received platinum-based chemotherapy only
* Age \>= 18 years
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2
* Leukocytes \>= 3000/mcL (performed within 14 days prior to registration)
* Absolute neutrophil count (ANC) \>= 1500/mcL (performed within 14 days prior to registration)
* Platelets \>= 100 000/mcL (performed within 14 days prior to registration)
* Hemoglobin \>= 9 g/dL or \>= 4.9 mmol/L (performed within 14 days prior to registration)
* Creatinine =\< 1.5 x upper limit of normal (ULN) OR measured or calculated creatinine clearance (CrCl) \>= 30 mL/min for patient with creatinine levels \> 1.5 x institutional ULN (performed within 14 days prior to registration)

  * Creatinine clearance (CrCl) should be calculated per institutional standard
  * Glomerular filtration rate (GFR) can also be used in place of creatinine or CrCl
* Total bilirubin =\< 1.5 x ULN OR direct bilirubin =\< ULN for patients with total bilirubin levels \> 1.5 x ULN (performed within 14 days prior to registration)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x ULN OR =\< 5 x ULN for patients with liver metastases (performed within 14 days prior to registration)
* International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 x ULN unless patient is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants (performed within 14 days prior to registration)
* Activated partial thromboplastin time (aPTT) =\< 1.5 x ULN unless patient is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants (performed within 14 days prior to registration)
* The effects of pembrolizumab (MK-3475) and tazemetostat on the developing human fetus are unknown. For this reason and because PD-1 inhibitors as well as EZH2 inhibitors are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation

  * Female patients of childbearing potential must have a negative urine or serum pregnancy test within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required
  * Female patients of childbearing/reproductive potential must be willing to use an adequate method of contraception, for the course of the study through 6 months for females and 3 months for males, respectively, after the last dose of study medication. Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the patient
  * Male patients of reproductive potential must agree to use an adequate method of contraception, starting with the first dose of study therapy through 3 months after the last dose of study therapy. Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the patient
  * Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately
* Participants who have the ability to understand and the willingness to sign an Institutional Review Board (IRB) approved written informed consent document are eligible OR Participants with impaired decision-making capacity (IDMC) who have a legally authorized representative (LAR) or caregiver are eligible
* Human immunodeficiency virus (HIV)-infected patients who do not have a history of Kaposi sarcoma and/or Multicentric Castleman Disease, who are on effective anti-retroviral therapy, and who have undetectable viral load within 6 months are eligible for this trial

Exclusion Criteria:

* Patients with disease that is suitable for local therapy administered with curative intent are not eligible
* Patients who have had chemotherapy, targeted small molecule therapy, or radiotherapy within 4 weeks prior to entering the study are not eligible
* Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1 per Common Terminology Criteria for Adverse Events \[CTCAE\] v.5 ) are not eligible

  * Note: patients with =\< grade 2 neuropathy or =\< grade 2 alopecia or =\< grade 3 audiometric hearing loss are an exception to this criterion and may qualify for the study
  * Note: if patients received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy
* Patients are not eligible who are currently participating and receiving study therapy or have participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment
* Patients who have received transfusion of blood products (including platelets or red blood cells) or administration of colony stimulating factors (including granulocyte colony-stimulating factor \[G-CSF\], granulocyte macrophage colony-stimulating factor \[GM-CSF\], or recombinant erythropoietin) within 4 weeks prior to the first dose of treatment are not eligible
* Patients with a diagnosis of immunodeficiency or are receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment are not eligible

  * Note: the use of physiologic doses of corticosteroids may be approved after consultation with the study principal investigator (PI)
* Patients with thrombocytopenia, neutropenia, or anemia of grade 3 (per CTCAE 5.0 criteria) are not eligible
* Patients with abnormalities known to be associated with MDS (e.g. del 5q, chr 7 abn) and myeloproliferative neoplasms (MPN, e.g. JAK2 V617F) observed in cytogenetic testing and deoxyribonucleic acid (DNA) sequencing are not eligible
* Patients with an ongoing or untreated hematologic malignancy or myeloproliferative disorder, or a prior history of a hematologic malignancy or myeloproliferative disorder are not eligible. (Examples of excluded malignancies/disorders include but are not limited to myelodysplastic syndrome \[MDS\], T cell lymphoblastic lymphoma (T-LBL), T cell acute lymphoblastic leukemia (T-ALL), and any other myeloid or lymphoid malignancy)
* Patients who have received prior PD-L1/PD-1/PD-L2 or EZH2 inhibitor therapy are not eligible
* Patients who have had a prior monoclonal antibody within 4 weeks prior to study day 1 are not eligible
* Patients must be disease-free of prior invasive malignancies for \> 5 years, with the exception of curatively-treated basal cell or squamous cell carcinoma of the skin or carcinoma in situ of the cervix. If there is a history of prior malignancy, patients must not be receiving other specific treatment for that cancer
* Patients with known brain metastases or carcinomatous meningitis are excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events

  * Note: patients with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging using the identical imaging modality for each assessment, either magnetic resonance imaging (MRI) or computed tomography (CT) scan, for at least 4 weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment
* Patients with a history of allergic reactions attributed to compounds of similar chemical or biologic composition to pembrolizumab (MK-3475) or tazemetostat are not eligible
* Patients with an active autoimmune disease that has required systemic treatment in the past 2 years (i.e., with use of disease modifying agents, corticosteroids, or immunosuppressive drugs) are not eligible

  * Note: replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment
* Patients with a history of (non-infectious) pneumonitis that required steroids or current pneumonitis are not eligible
* Patients with a prolongation of corrected QT interval (Fridericia's correction formula \[QTcF\]) of \> 450 msec are not eligible
* Patients with major surgery within 3 weeks before the first dose of study drugs

  * Note: minor surgery (e.g. minor biopsy of an extracranial site, central venous catheter placement, shunt revision) has no restriction
* Patients must not have a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the patient's participation for the full duration of the trial, or is not in the best interest of the patient to participate, in the opinion of the treating investigator
* Patients receiving any medications or substances that are strong or moderate inhibitors or inducers of CYP3A =\< 14 days prior to study treatment are not eligible

  * Note: The study team should check a frequently-updated medical reference for a list of drugs to avoid or minimize use of. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product
* Patients who are unable to take oral medication OR have malabsorption syndrome or any other uncontrolled gastrointestinal condition (e.g., nausea, diarrhea, vomiting) that might impair the bioavailability of tazemetostat are not eligible
* Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, interstitial lung disease or active, non-infectious pneumonitis, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, uncontrolled arterial hypertension, stroke within 6 months prior to starting study treatment, or psychiatric illness/social situations that would limit compliance with study requirements are not eligible
* Pregnant women are excluded from this study because pembrolizumab (MK-3475) and tazemetostat are agents with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with pembrolizumab (MK-3475) and tazemetostat, breastfeeding should be discontinued if the mother is treated with pembrolizumab (MK-3475) or tazemetostat

  * Pembrolizumab (MK-3475) may have adverse effects on a fetus in utero. Furthermore, it is not known if pembrolizumab (MK-3475) has transient adverse effects on the composition of sperm
  * Patients are excluded from this study if pregnant or breastfeeding, or expecting to conceive or father children (or donate sperm) within the projected duration of the trial, starting with the screening visit through 6 months after the last dose of trial treatment
* Patients with a known history of hepatitis B (defined as hepatitis B surface antigen \[HBsAg\] reactive) or known hepatitis C virus (defined as HCV ribonucleic acid \[RNA\] \[qualitative\] is detected) infection are not eligible

  * Note: no testing for hepatitis B and hepatitis C is required unless mandated by local health authority
* Patients who have received a live vaccine within 30 days of planned treatment start are not eligible

  * Note: seasonal flu vaccines that do not contain live virus are permitted