Testing Drug Treatments After CAR T-cell Therapy in Patients With Relapsed/Refractory Diffuse Large B-cell Lymphoma

Inclusion Criteria: * STEP 1: REGISTRATION: Participants must have a histologically confirmed diagnosis of diffuse large B-cell lymphoma or follicular lymphoma grade 3b or primary mediastinal large B-cell lymphoma (PMBCL) * STEP 1: REGISTRATION: Participants with transformed DLBCL must have transformed DLBCL from follicular or marginal zone lymphoma * STEP 1: REGISTRATION: Participant must have bi-dimensionally measurable systemic disease (at least one lesion with longest diameter \> 1.5 cm) * STEP 1: REGISTRATION: Participants with secondary central nervous system (CNS) lymphoma (parenchymal, spinal cord, meningeal, cerebrospinal fluid involvement) must be asymptomatic from their CNS disease * STEP 1: REGISTRATION: Participants must be registered for step 1 after they have signed institutional consent for CAR T-cell leukapheresis but prior to the start of lymphodepleting (LD) chemotherapy for commercial CAR T-cell product * STEP 1: REGISTRATION: In the opinion of the enrolling physician, participants must be felt to be a candidate for CAR T-cell therapy with plans to be treated with Food and Drug Administration (FDA) approved commercially available CD19 CAR T-cell construct. * Participants must qualify for commercially approved CD19 CAR T-cell therapy per FDA package insert. * If the CAR T-cell product does not meet parameters to be given as an FDA approved product (i.e. does not meet specification criteria mandated by FDA and is infused under an expanded access protocol \[EAP\] or single participant investigational new drug \[IND\]) the participant will be taken off of study and no longer be eligible for step 2 randomization * STEP 1: REGISTRATION: Participants are permitted to receive or have received 'bridging therapy' after CAR T-cell leukapheresis. However, participants must not receive polatuzumab vedotin, and/or mosunetuzumab as part of bridging therapy. * Bridging therapy is defined as lymphoma directed therapy administered between leukapheresis and the start of LD chemotherapy. This includes cytotoxic chemotherapy (e.g.: bendamustine and rituximab \[BR\], rituximab, gemcitabine and oxaliplatin \[R-gem/ox\]), radiation, corticosteroids, as well as novel therapies such as BTK inhibitors (e.g.: Ibrutinib), immunomodulators (e.g.: lenalidomide), monoclonal antibodies (e.g.: rituximab, obinutuzumab, tafasitamab) antibody drug conjugates (e.g: loncastuximab), checkpoint inhibitors (e.g.: pembrolizumab, nivolumab), clinical trial treatments, etc. * If a participant receives polatuzumab vedotin or mosunetuzumab as bridging they will ineligible to continue on step 1 registration portion of the study and be ineligible for step 2 randomization * STEP 1: REGISTRATION: PET-CT scan must be planned for completion within 60 days prior to the start of LD chemotherapy. * All pre-CAR T-cell therapy disease must be assessed and documented on the baseline/pre-registration tumor assessment form. * If receiving bridging therapy, participants must have a PET-CT scan upon completion of all planned bridging therapy. If the PET-CT scan after completion of bridging therapy is consistent with complete remission per Lugano criteria as determined by enrolling physician, that participant will be ineligible for step 2 randomization. * Participants are permitted to receive corticosteroids after leukapheresis without the need to repeat a PET-CT scan. If steroids are used, they must be planned to stop no later than 3 days before CAR -T cell infusion. * If response assessment by central review cannot be completed (I.e., poor quality of PET-CT scan, PET-CT performed out of window, etc.) this would be recorded as 'inadequate assessment' and patient would not be eligible for randomization * STEP 1: REGISTRATION: Participants that have previously been treated with polatuzumab vedotin or mosunetuzumab prior to CAR T-cell leukapheresis for either indolent or aggressive NHL are eligible as long as the participant did not have refractory disease or progression/relapse within 6 months of the last infusion with either agent * STEP 1: REGISTRATION: Participants must be planning to receive CAR T-cell infusion no earlier than 2 days and no later than 14 days after completion of the last day of lymphodepleting chemotherapy. Any participant receiving CAR T-cell infusion outside of this window will be ineligible for step 2 randomization * STEP 1: REGISTRATION: LD chemotherapy prior to CAR T-cell infusion must be planned to start within 60 days after step 1 registration * STEP 1: REGISTRATION: Participants must be \>= 18 years of age at the time of registration * STEP 1: REGISTRATION: Participants must have Zubrod performance score (PS) of 0, 1, or 2 * STEP 1: REGISTRATION: Total bilirubin =\< 2 x institutional upper limit of normal (ULN) (within 14 days prior to registration) * Unless due to Gilbert's disease or lymphomatous involvement of liver * STEP 1: REGISTRATION: Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 3 x institutional ULN (within 14 days prior to registration) * STEP 1: REGISTRATION: Creatinine clearance \>= 40 mL/min, as estimated by the Cockcroft and Gault formula. The creatinine value used in the calculation must have been obtained within 14 days prior to registration. Estimated creatinine clearance is based on actual body weight * STEP 1: REGISTRATION: Participants must have an echocardiogram (ECHO) or multigated acquisition scan (MUGA) within 60 days prior to registration with a cardiac ejection fraction \>= 40%. * Participants with current symptoms of cardiac disease must have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, participants must be class 2B or better. * Participants must not have documented myocardial infarction and percutaneous coronary intervention (PCI) within 6 months prior to registration or myocardial infarction without PCI within 3 months of reg