Study to Evaluate Adverse Events and Change in Disease Activity When Intravenously (IV) Infused Livmoniplimab is Used in Combination With IV Infused Budigalimab in Adult Participants With Urothelial Carcinoma (UC)

Inclusion Criteria:

* Participant has histologically or cytologically confirmed urothelial carcinoma (i.e., cancer of the bladder, renal pelvis, ureter, or urethra). Mixed histologic types are allowed if urothelial (transitional cell) is the predominant histology.
* Participant has radiologically documented metastatic disease.
* Participant must have experienced radiographic progression or relapse on checkpoint inhibitor (anti-programmed cell death protein 1 \[PD-1\] or anti-programmed death-ligand 1 \[PD-L1\]) in the metastatic, adjuvant, or neo-adjuvant setting. Participant must have received at least 2 cycles of anti-PD-1 or anti-PD-L1.
* Participants eligible for platinum must have received a platinum containing regimen (cisplatin or carboplatin) in the metastatic, locally advanced, neoadjuvant, or adjuvant setting. If platinum was administered in the neoadjuvant or adjuvant setting, participant must have progressed within 6 months of completion of treatment. Platinum ineligible participants may enroll in this study without receiving a platinum containing regimen.
* Participant has at least 1 measurable lesion per response evaluation criteria in solid tumors (RECIST) v1.1 as determined by investigator.
* Life expectancy must be at least 3 months.

Exclusion Criteria:

* Participant has received more than 1 prior chemotherapy regimen for urothelial cancer in metastatic setting, including chemotherapy agents planned in comparator arm.

  * Platinum based chemotherapy administered in adjuvant or neoadjuvant setting will count towards this criterion if participant progressed within 6 months of completion.
  * Chemotherapy administered during concurrent chemoradiotherapy for primary cancer will not count towards this criterion.
  * The substitution of carboplatin for cisplatin does not constitute a new regimen provided no new chemotherapeutic agents were added to the regimen and no progression was noted prior to the change in platinum.
  * Antibody-drug conjugate (ADC) will not count towards this criterion.
  * Participant who previously received gemcitabine in combination with platinum in metastatic setting will be eligible to receive docetaxel or paclitaxel in comparator arm.
* Participant has received more than 1 antibody-drug conjugate (ADC) in metastatic setting.
* Has had prior radiation therapy within 28 days prior to first dose of study drug or who has not recovered (i.e., \<= Grade 1 or at baseline) from adverse events due to radiotherapy.
* History of additional malignancy or history of prior malignancy, except for adequately treated basal or squamous skin cancer, or cervical carcinoma in situ without evidence of disease, or malignancy treated with curative intent and with no evidence of disease recurrence for 5 years since the initiation of that therapy.
* Prior allogeneic stem cell or solid organ transplantation.