A Phase II Trial to Evaluate Pemetrexed Response in Relation to Tumor Alterations of Gene Status in Patients With Previously Treated Metastatic Urothelial Carcinoma and Other Solid Tumors

Inclusion Criteria: * Patients must have pathologically or cytologically confirmed metastatic urothelial bladder carcinoma (Arm A) or other metastatic/locally-advanced solid malignancy (Arm B) and MLL4- protein (KMT2D-gene) or UTX-protein (KDM6A-gene) or MTAP loss of function mutation including but not limited to SNVs that cause truncation, CNVs, and indels confirmed by next generation sequencing or immunohistochemistry techniques. * Patients must have at least 1 measurable lesion per Response Evaluation Criteria in Solid Tumors version 1.1 (RECIST v1.1), measured preferably by computed tomography (CT) scan. * Patients who have received any prior neoadjuvant or systemic chemotherapy are eligible. * Notes: * Patients must have progressive disease despite two prior lines of therapy in the metastatic setting or locally advanced setting unless the patient was not suitable for an approved second line regimen due to intolerance or another clinical factor; * Treatment cannot have included prior pemetrexed. Any prior intravesical therapy, or immunotherapy is allowed. At least 3 weeks (21 days) wash-out period since prior chemotherapy or radiation therapy or targeted agent is required. * Patients must be aged ≥ 18 years. * Patients must exhibit an Eastern Cooperative Oncology Group (ECOG) performance status of 0-2. * Patients must have adequate organ and bone marrow function as defined: * Absolute neutrophil count (ANC) ≥ 1,500/mcL (growth factor allowed and can be added at the discretion of the treating oncologist) * Hemoglobin (Hgb) ≥ 8.5 g/dL (without the need for transfusion within the previous one week) * Platelets (PLT) ≥ 100,000/mL (without the need for platelet transfusion within the previous one week) * Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN), except subjects with Gilbert's syndrome or liver metastases, who must have a baseline total bilirubin ≤ 3.0 mg/dL * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) ≤ 3 x institutional ULN or ≤ 5 x ULN if documented liver metastases are present * Alanine aminotransferase (ALT) (serum glutamic-pyruvic transaminase \[SGPT\]) ≤ 3 x institutional ULN or ≤ 5 x ULN if documented liver metastases are present * Creatinine clearance ≥ 45 mL/min/1.73 m\^2 using the standard Cockcroft and Gault formula * Patients must have the ability to comply with the administration of supplemental therapies including folic acid, vitamin B12 and steroids as directed by study team and as per standard of care and institutional standards and practice for pemetrexed use. * Patients must be able swallow oral medication or not have problems/diseases that affect absorption or oral medication. * Patients with a known history of human immunodeficiency virus (HIV), infected patients on effective anti-retroviral therapy must have a viral load undetectable for 6 months prior to registration. Please note this lab is not a requirement for eligibility, however, if it was previously done as part of the patient's health care, it should be documented for eligibility. * Patients with a known history of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. Please note this lab is not a requirement for eligibility, however, if the lab has been completed previously as part of the patient's health care, then it should be documented for eligibility. * Patients with a known history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with a known HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. Please note this lab is not a requirement for eligibility, however if it was previously done as part of the patient's health care, it should be documented for eligibility. * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression. * Pemetrexed is known to be teratogenic. For this reason, patients of child-bearing potential (POCBP) and their partners with sperm-producing reproductive capacity must agree to use adequate contraception from time of informed consent, for the duration of study participation, and for 180 days following completion of pemetrexed therapy. Should a POCBP become pregnant or suspect they are pregnant while they or their partner are participating in this study, they should inform their treating physician immediately. Patients with sperm-producing reproductive capacity (PWSPRC) treated or enrolled on this protocol must also agree to use adequate contraception with partners of childbearing potential from time of informed consent, for the duration of study participation, and 180 days after completion of administration * Note: A POCBP is any patient (regardless of gender, sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) with an egg-producing reproductive tract who meets the following criteria: * Has not undergone a hysterectomy or bilateral oophorectomy * Has had menses at any time in the preceding 12 consecutive months (and therefore has not been naturally postmenopausal for \> 12 months) * POCBP must have a negative pregnancy test prior to registration on study. * The ability to interrupt nonsteroidal anti-inflammatory drugs (NSAIDS) or aspirin at higher dose (\> 1.3 g per day) 2 days before (5 days for long-acting NSAIDs), the day of, and 2 days following administration of pemetrexed. * Patients must be able to understand and voluntarily sign a written informed consent and willing and able to comply with the protocol requirements including scheduled visits, treatment plan, laboratory tests and other study procedures. Exclusion Criteria: * Patients who received prior pemetrexed containing chemotherapy. * Patients who have had chemotherapy or radiotherapy ≤ 21 days (prior to planned treatment start date). For palliative radiation, 5 day wash out is sufficient. * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia, neuropathy and other non-significant adverse events deemed not clinically significant by the treating investigator, adverse events per National Cancer Institute Common Terminology Criteria for Adverse Events version 5.0 (NCI CTCAE v 5.0). * Patients who are receiving any other investigational agents. A 21 day wash out period will be required after discontinuation of an investigational agent prior to first day of study treatment. * Patients who have a history of allergic reactions attributed to compounds of similar chemical or biologic composition to pemetrexed. * Patients who have an uncontrolled intercurrent illness including, but not limited to any of the following: * Ongoing or active infection requiring systemic treatment * Symptomatic congestive heart failure * Unstable angina pectoris * Any other illness or condition that the treating investigator feels would interfere with study compliance or would compromise the patient's safety or study endpoints * Patients with a prior or concurrent malignancy whose natural history or treatment has the potential to interfere with the safety or efficacy assessment of the investigational regimen. * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. * Note: To be eligible for this trial, patients should be class 2B or better * Patients with presence of third space fluid which cannot be controlled by drainage * Note: For patients who develop or have baseline clinically significant pleural or peritoneal effusions (on the basis of symptoms or clinical examination) before or during initiation of pemetrexed therapy, consideration should be given to draining the effusion prior to dosing. However, if, in the investigator's opinion, the effusion represents progression of disease, the patient should be discontinued from study therapy * Has received the final dose of any of the following treatments/ procedures with the specified minimum intervals before first dose of study drug: * Focal radiation therapy - 7 days * Surgery with general anesthesia - 7 days * Surgery with local anesthesia - 3 days * For palliative radiation - 5 days * Patients of child bearing (POCB) potential who are pregnant or nursing. * Note: Registration of patients is completed in Northwestern Oncology Trial Information System (NOTIS)