Inclusion Criteria:  Life expectancy of at least 12 weeks History of relapsed or refractory DLBCL, HGBCL, PMBCL or tFL who have relapsed after or failed to respond to at least two prior standard systemic treatment regimens that include at least one prior regimen containing an anthracycline and at least one containing an anti-CD20-directed therapy and for whom there is no available therapy expected to improve survival (e.g., standard chemotherapy, autologous or allogeneic stem cell transplant). PET/CT scan (preferred), diagnostic CT scan, or MRI prior to CAR-T cell therapy, with at least one bi-dimensionally measurable lesion (≥ 1.5 cm for nodal lesion or ≥ 1cm for extra-nodal lesions in largest dimension by low-dose computerized tomography [CT] scan with FDG-uptake ≥ liver); this imaging must have been obtained within 56 days of receiving CAR T cell therapy. PET/CT scan (preferred), diagnostic CT scan, or MRI with at least one bi-dimensionally measurable lesion (≥ 1.5 cm for nodal lesion or ≥ 1cm for extra-nodal lesions in largest dimension by low-dose computerized tomography [CT] scan with FDG-uptake ≥ liver); this imaging documenting measurable disease must be obtained at least day +30 after CAR T cell infusion and prior to cycle 1 day 1. Be at least 30 days after CAR T-cell infusion at time of study enrollment. Adequate laboratory studies, Ability and willingness to take proper contraceptive precautions  Exclusion Criteria:  Had > Grade 3 cytokine release syndrome (CRS) by ASTCT criteria32 after CAR-T therapy or who have unresolved CRS after CAR-T therapy Had ≥ grade 2 neurologic toxicity by ASTCT criteria after CAR-T therapy or who have active neurologic toxicity after CAR-T therapy Treated with axicabtagene ciloleucel Inability to comply with protocol-mandated hospitalization and activities restrictions in the investigators' decision Pregnant or lactating, or intending to become pregnant during the study or within 3 months after the last dose of bispecific antibody or 18 months of obinutuzumab, whichever comes later Prior solid organ transplantation History of autoimmune disease, including but not limited to myocarditis, pneumonitis, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjögren's syndrome, Guillain-Barré syndrome, multiple sclerosis, uveitis, vasculitis, or glomerulonephritis History of confirmed progressive multifocal leukoencephalopathy (PML) History of severe allergic or anaphylactic reactions to monoclonal antibody therapy (or recombinant antibody-related fusion proteins) History of other malignancy that could affect compliance with the protocol or interpretation of results Significant cardiovascular disease such as New York Heart Association Class III or IV cardiac disease, myocardial infarction within the last 6 months, unstable arrhythmias, or unstable angina) Significant active pulmonary disease (e.g., bronchospasm and/or obstructive pulmonary disease) requiring oxygen or corticosteroid use. Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment, or any major documented infection requiring treatment with IV antibiotics or hospitalization within 2 weeks prior to first mosunetuzumab or glofitamab administration. Empiric or prophylactic antibiotics administered during neutropenia or neutropenic fever without microbiologic evidence of infection do not exclude patients. Recent major surgery within 4 weeks prior to first mosunetuzumab or glofitamab administration Active or chronic infection(s) would have increased risks for toxicity if treated with bispecific antibody therapy, thus will be excluded. Administration of a live, attenuated vaccine within 4 weeks before first dose of study treatment or anticipation that such a live attenuated vaccine will be required during the study Received systemic immunosuppressive medications (including but not limited to cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor agents) with the exception of corticosteroid treatment < 20 mg/day prednisone or equivalent within 2 weeks prior to first dose of bispecific antibody History of drug or alcohol abuse within 12 months prior to screening in the investigator's judgment Any serious medical condition or abnormality in clinical laboratory tests that, in the investigator's and/or Medical Monitor's judgment, precludes the patient's safe participation in and completion of the study, or which could affect compliance with the protocol or interpretation of results