This phase II trial tests whether mosunetuzumab and/or polatuzumab vedotin helps benefit patients who have received chemotherapy (fludarabine and cyclophosphamide) followed by chimeric antigen receptor (CAR) T-cell therapy (tisagenlecleucel, axicabtagene ciloleucel, or lisocabtagene maraleucel) for diffuse large B-cell lymphoma that has come back (recurrent) or that does not respond to treatment (refractory) or grade IIIb follicular lymphoma. Mosunetuzumab is a monoclonal antibody that may interfere with the ability of cancer cells to grow and spread. Polatuzumab vedotin is a monoclonal antibody, called polatuzumab, linked to a drug called vedotin. Polatuzumab is a form of targeted therapy because it attaches to specific molecules (receptors) on the surface of cancer cells, and delivers vedotin to kill them. Chemotherapy drugs, such as fludarabine and cyclophosphamide, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. CAR T-cell therapy is a type of treatment in which a patient's T cells (a type of immune system cell) are changed in the laboratory so they will attack cancer cells. T cells are taken from a patient's blood. Then the gene for a special receptor that binds to a certain protein on the patient's cancer cells is added to the T cells in the laboratory. The special receptor is called a chimeric antigen receptor. Large numbers of the CAR T cells are grown in the laboratory and given to the patient by infusion for treatment of certain cancers. Giving mosunetuzumab and/or polatuzumab vedotin after chemotherapy and CAR T-cell therapy may be more effective at controlling or shrinking the cancer than not giving them.