Cisplatin and Gemcitabine Hydrochloride With or Without Berzosertib in Treating Patients With Metastatic Urothelial Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed metastatic urothelial carcinoma; urothelial cancer derived from the bladder, ureter or upper tract is permitted * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) with conventional techniques or as \>= 10 mm (\>= 1 cm) with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Patients must have access to archival tumor tissue for proposed correlative studies; these may be derived from transurethral resection of bladder tumors (TURBT), cystectomy, or biopsy; if archival tissue is not available for proposed correlatives, patients may be enrolled at the discretion of the study principal investigator (PI) (SKP) * No prior cytotoxic chemotherapy for metastatic disease; prior immunotherapy is permitted * At least 12 months have elapsed since platinum-based peri-operative treatment * Karnofsky \>= 70% (Eastern Cooperative Oncology Group \[ECOG\] performance status 0-1) * Life expectancy of greater than 3 months * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine amin…