A Phase II Study of Bintrafusp Alfa (M7824) in Checkpoint Inhibitor Naive and Refractory Subjects With Urothelial Carcinoma

* INCLUSION CRITERIA: * Ability to understand the purpose of the study, provide signed and dated informed consent, and able to comply with all procedures. * Male or female patients aged greater than or equal to 18 years of age at time of consent. * Patients with histologically confirmed diagnosis of urothelial carcinoma of the urinary tract, including the renal pelvis, ureter, bladder, or urethra. Differentiation with variant histologies (e.g. squamous cell differentiated) will be permitted. Mixed histologies are required to have a dominant urothelial/transitional cell pattern. * Patients must have metastatic disease defined as new or progressive lesions on cross- sectional imaging. Radiological evaluation should occur within 21 days prior to enrollment. * Patient must have evaluable and measurable disease, per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. * Patients may have been previously treated with prior cytotoxic chemotherapy regimen or targeted agent. Patients may have received any number of prior cytotoxic agents. * Patients may have had prior immunomodulating therapy including therapy targeting the Programmed death-1 (PD-1)/Programmed Cell Death Ligand 1 (PD-L1 axis (cohort 2A and B) but excluding prior treatment with Bintrafusp Alfa (M7824). * Pre-treatment tissue biopsy and/or archival tissue availability for PD-L1 expression testing is mandatory for enrollment. * Eastern Cooperative Oncology Group (ECOG) performance status less than or equal to 2 * Required laboratory values reflective of organ function are listed below: * Absolute neutrophil count greater than or equal to 1000/microliter * Platelets greater than or equal to 75,000 microliter * Hemoglobin greater than or equal to 9 g/dL (erythrocyte transfusions are allowed to achieve acceptable Hgb) * Aspartate aminotransferase (AST)/Serum glutamic oxaloacetic transaminase (SGOT)/Alanine aminotransferase (ALT)/Serum glutamic-pyruvic transaminase (SGPT) less than or equal to 1.5 institutional upper limit of normal (ULN) with the following exception: ---Patients with liver involvement who have AST and ALT less than or equal to 5 ULN may be enrolled. * Total bilirubin within normal limits with the following exceptions: * Patients with known Gilbert disease who have serum bilirubin level less than or equal to 3 ULN may be enrolled. * Patients with tumor liver involvement bilirubin with less than or equal to 3.0 ULN. * International normalized ratio (INR) and activated partial thromboplastin time (aPTT) less than or equal to 1.5 ULN ---This applies only to patients who are not receiving therapeutic anticoagulation; patients receiving therapeutic anticoagulation (such as low-molecular-weight heparin or warfarin) should be on a stable dose. * Creatinine clearance (CrCl) greater than or equal to 30 mL/min/1.73 m\^2 (glomerular filtration rate (GFR) may be used in place of CrCl. Creatinine clearance or estimated glomerular filtration rate (eGFR) should be calculated per institutional standard) * The effects of M7824 on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use strict and effective contraception (hormonal or barrier method of birth control; abstinence) during treatment and for at least 65 days for women and 125 days for men, after the last dose of M7824 administration. 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. * Human immunodeficiency virus (HIV) positive patients are eligible if on stable dose of highly active antiretroviral therapy (HAART), cluster of differentiation 4 (CD4) counts are greater than 350 cells/mm3 and viral load is undetectable. * Patients with previously treated brain or central nervous system (CNS) metastases are eligible provided that the subjects have recovered from any acute effects of radiotherapy and is not requiring steroids, and any whole brain radiation therapy or any stereotactic radiosurgery was completed at least 2 weeks prior to M7824 administration. * Hepatitis B virus (HBV) positive patients are eligible-they must have been treated and on a stable dose of antivirals (eg, entecavir, tenofovir, or lamivudine; adefovir or interferon are not allowed) at study entry and with planned monitoring and management according to appropriate labeling guidance. * Hepatitis C virus (HCV) positive patients are eligible if participants are on active HCV therapy at study entry and on a stable dose without documented clinically significant impaired liver function test or hematologic abnormalities and with planned monitoring and management according to appropriate labeling guidance. * Cohort 1A Cisplatin Ineligible Specific Inclusion Criteria (first-line for metastatic cisplatin-ineligible): * No prior chemotherapy for inoperable locally advanced or metastatic or recurrent Urothelial Carcinoma (UC) ---For patients who received prior adjuvant/neoadjuvant chemotherapy or chemoradiation for UC, a treatment-free interval \> 12 months between the last treatment administration and the date of recurrence is required in order to be considered treatment naive in the metastatic setting. Prior local intravesical chemotherapy or immunotherapy is allowed if completed at least 4 weeks prior to the initiation of study treatment. * Ineligible ("unfit") for chemotherapy or cisplatin-based chemotherapy as defined by any one of the following criteria: * Impaired renal function (CrCl \> 30 but \< 60 mL/min); GFR should be calculated per institutional standard. * A hearing loss (measured by audiometry) of 25 decibels (dB) at two contiguous frequencies * Grade greater than or equal to 2 peripheral neuropathy (i.e., sensory alteration or paresthesias including tingling) * Eastern Cooperative Oncology Group (ECOG) performance score of 2 * Patient declines chemotherapy after informed discussion with the study doctor * Cohort 1B Refractory Post-platinum Therapy Specific Inclusion Criteria (second-line for metastatic disease): --Disease progression during or following treatment with a platinum-containing regimen for inoperable locally advanced or metastatic urothelial carcinoma or disease recurrence. Examples of regimens include cisplatin + gemcitabine (GC), methotrexate + vinblastine sulfate + doxorubicin + cisplatin (MVAC), and carboplatin + gemcitabine (CarboGem). ---Patients who received prior adjuvant/neoadjuvant chemotherapy and progressed within 12 months of treatment with a platinum-containing adjuvant/neoadjuvant regimen will be considered as second-line patients. * Cohort 2A Checkpoint Inhibitor Previously Treated Patients that Previously Achieved a Complete Response (CR) or Partial Response (PR) Specific Inclusion Criteria: --Patients must have been treated with at least one treatment of a PD-1/PD-L1 checkpoint inhibitor for advance or metastatic UC and achieved a complete response or partial response by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria. * Cohort 2B Checkpoint inhibitor previously treated patients that previously had stable disease (SD) or progressive disease (PD) * Specific Inclusion Criteria: * Patients must have been treated with at least one treatment of a PD-1/PD-L1 checkpoint inhibitor for advance or metastatic UC and had stable disease or a progressive disease by RECIST 1.1 criteria. EXCLUSION CRITERIA: * History of allergic reactions attributed to compounds of similar chemical or biologic composition to M7824 investigational agents used in the study. * Uncontrolled intercurrent illness including, but not limited to, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Symptomatic central nervous system metastasis. * Subjects unwilling to accept blood products as medically indicated * Pregnant women are excluded from this study because M7824 is an agent 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 M7824, breastfeeding should be discontinued if the mother is treated with these agents. * Patients with any active or recent history of a known or suspected autoimmune disease (with the exception of diabetes type I, vitiligo, psoriasis, or hypo- or hyperthyroid diseases not requiring immunosuppressive treatment) or recent history of a syndrome that required treatment with either systemic corticosteroids (\>10 mg daily prednisone equivalent) or immunosuppressive medications. Inhaled steroids and adrenal replacement steroid doses up to 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. * Patients with a currently active second malignancy other than non-melanoma skin cancers or cervical carcinoma in situ or incidental organ-confined prostate cancer found on cystoprostatectomy (provided that the following criteria are met: Stage T2N0M0 or lower; Gleason score \<= 3+4, prostate-specific antigen (PSA) undetectable). Patients are not considered to have a currently active malignancy if they have completed therapy and are free of disease for \>= 2 years and currently do not require systemic therapy. * Patients who have received or will receive a live vaccine within 30 days prior to the first administration of study intervention. Seasonal flu vaccines that do not contain a live virus are permitted. Locally approved COVID vaccines are permitted. * Patients having tumor lesion(s) in the liver or chest which are 10 cm or larger. * Patients previously treated with M7824. * Patients previously treated with PD-1/PD-L1 checkpoint inhibitors (for Cohorts 1A and 1B only)