Avelumab in Combination With Fluorouracil and Mitomycin or Cisplatin and Radiation Therapy in Treating Participants With Muscle-Invasive Bladder Cancer

PRIMARY OBJECTIVES: I. To evaluate the complete response rate of concurrent chemotherapy radiation treatment combined with avelumab for patients with muscle invasive bladder cancer. SECONDARY OBJECTIVES: I. To evaluate the safety and toxicity (adverse event profile) of concurrent chemotherapy radiation treatment combined with avelumab. II. To evaluate quality of life (QoL) at 1 year of concurrent chemotherapy radiation treatment combined with avelumab. III. To evaluate progression-free survival and relapse-free survival at 1 year with concurrent chemotherapy radiation treatment combined with avelumab. CORRELATIVE OBJECTIVES: I. To explore biomarkers that may predict response to avelumab in the muscle invasive population. II. To evaluate the association of tumor mutational burden with response to concurrent chemo- radiation and immunotherapy. III. To evaluate whether concurrent chemoradiation and immunotherapy after maximal transurethral resection of bladder tumor (TURBT) is associated with a decrease in circulating Bim+CD11a\^high PD-1+CD8+ T-cells and myeloid-derived suppressor cells (MDSCs). OUTLINE: Participants receive avelumab intravenously (IV) over 60 minutes every 14 days for a total of 10 courses in the absence of disease progression or unacceptable toxicity. Beginning 29 days after the first dose of avelumab, participants receive either fluorouracil IV on days 1-5 and 16-20 during radiation therapy (RT) and mitomycin IV on day 1 of course 3, or cisplatin IV starting on day 1 of courses 3-5 for up to 6 weeks in the absence of disease progression or unacceptable toxicity. After completion of study treatment, participants are followed up at 30 days, 6, 9, and 12 months.