Pembrolizumab With Combination Chemotherapy in Treating Participants With Locally Advanced or Metastatic Small Cell/Neuroendocrine Cancers of Urothelium or Prostate

Inclusion Criteria: * Histologically confirmed diagnosis of locally advanced or metastatic 1) naive small cell cancer of the bladder, urethra, or upper urinary tract, or 2) primary small cell or neuroendocrine prostate cancer will be enrolled in this study. * Histological diagnosis of pure or mixed small cell or neuroendocrine cancer by a genitourinary pathologist is sufficient and confirmatory immunohistochemistry is not required. * Cohort 1 will include subjects with no prior systemic chemotherapy for locally advanced or metastatic urothelial carcinoma, with the following exception(s): * Platinum-based chemotherapy with recurrence \> 12 months from completion of therapy is permitted. * Cohort 2 will include subjects with no prior systemic chemotherapy for primary small cell prostate cancer, with the following exception(s): * Platinum-based chemotherapy with recurrence \> 12 months from completion of therapy is permitted. * Cohort 2 will include subjects with prior treatments for metastatic castration-resistant prostate cancer (mCRPC) including: * Prior chemotherapy with 2 other agents is allowed if \> 6 months elapsed from last dose (if docetaxel chemotherapy is used more than once for hormone-sensitive and for mCRPC it will be considered 1 therapy). * Ongoing androgen deprivation therapy with up to 2 second-generation hormonal manipulations (e.g. including but not limited to abiraterone acetate and/or enzalutamide). * Ongoing treatment with for bone metastasi…