• Signed informed consent.

• Eastern Collaborative Oncology Group (ECOG) performance status score of 0 or 1.

• Body weight \> 30 kg.

• Absolute neutrophil count (ANC) \>= 1500 mm\^3 (within 28 days before the first study treatment).

• Hemoglobin \>= 9.0 g/dL (within 28 days before the first study treatment).

• Platelet count \>= 100,000 per mm\^3 (within 28 days before the first study treatment).

• Serum bilirubin =\< 1.5 X upper limit of normal (ULN) (within 28 days before the first study treatment). Subjects with Gilbert's syndrome will be considered after consultation with the principal investigator (PI).

• Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 X ULN (within 28 days before the first study treatment).

• For subjects who will be treated with dose dense methotrexate, vinblastine, doxorubicin, and cisplatin (DD MVAC) or cisplatin and gemcitabine (CG), creatinine clearance \>= 50 mL/min as measured based on Cockcroft-Gault glomerular filtration rate estimation (within 28 days before the first study treatment).

• For subjects who will be treated with carboplatin and gemcitabine (Carbo Gem), creatinine clearance \>= 30 mL/min as measured based on Cockcroft-Gault glomerular filtration rate estimation (within 28 days before the first study treatment).

• Anticipated life expectancy of \>= 12 weeks as assessed by the investigator.

• Histologically proven carcinoma of the bladder of variant urothelial carcinoma histologies which include squamous, adenocarcinoma, nested, plasmacytoid, micropapillary, glandular differentiation, lipid cell, clear cell, undifferentiated, giant cell, trophoblastic, sarcomatoid, carcinosarcoma; subjects with mixed cell types are eligible.

• Clinical T stage 2 (cT2) T4a, N0 N1, M0 disease. Clinical T stage is based on the transurethral resection of bladder tumor (TURBT) sample and imaging studies. Subjects must undergo cystoscopy and TURBT as part of screening within 30 days prior to registration.

• Abdominal/pelvic imaging by computed tomography (CT) or magnetic resonance imaging (MRI) scan; chest imaging by CT scan or x-ray (CT/positron emission tomography (PET) within 30 days prior to registration.

• Resting 12 lead electrocardiogram (ECG) documenting Fridericia's correction formula (QTcF) =\< 470 ms.

• Consent to provide a formalin fixed paraffin embedded (FFPE) tissue block and 1 hematoxylin and eosin (H and E) slide (preferred) or one of the following:

‣ 10 unstained slides and 1 H and E slide OR

⁃ Tissue block punches and 1 H and E slide, OR

⁃ 4 to 6 cores and 1 H and E slide.

• Willing and able to comply with the protocol for the duration of the study including undergoing treatment, scheduled visits and examinations including follow up.

• For female subjects:

‣ Women \< 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of exogenous hormonal treatments and if they have luteinizing hormone and follicle stimulating hormone levels in the post-menopausal range for the institution or underwent surgical sterilization (bilateral oophorectomy or hysterectomy).

⁃ Women \>= 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of all exogenous hormonal treatments, had radiation induced menopause with last menses \> 1 year ago, had chemotherapy induced menopause with last menses \> 1 year ago, or underwent surgical sterilization (bilateral oophorectomy, bilateral salpingectomy or hysterectomy).