Inclusion Criteria:  Age >/= 18 years Eastern Cooperative Oncology Group (ECOG) Performance Status of 0, 1, or 2 History of one of the following histologically documented hematologic malignancies that are expected to express the CD20 antigen: For Dose-escalation: Grades 1-3a follicular lymphoma (FL), transformed indolent non-Hodgkin lymphoma (NHL), diffuse large B-cell lymphoma (DLBCL); For Dose expansion: Grades 1-3a FL and 1L (treatment naive) FL, DLBCL/transformed FL Fluorodeoxyglucose-avid lymphoma (i.e. PET-positive lymphoma) At least one bi-dimensionally measurable nodal lesion (> 1.5 cm in its largest dimension by PET/CT scan), or at least one bi-dimensionally measurable extranodal lesion (> 1.0 cm in its largest dimension by PET/CT scan) Availability of a representative tumor specimen and the corresponding pathology report for confirmation of the diagnosis of NHL A fresh pretreatment biopsy during screening period, excisional or incisional, is preferred Adequate hematologic function Normal laboratory values All participants and health care providers will be trained and counseled on pregnancy prevention. For female participants of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraception during the treatment period and for 3 months after the final dose of mosunetuzumab, at least 18 months after pre-treatment with obinutuzumab or 2 months after the last dose of glofitamab, 28 days after the last dose of CC-220 and CC-99282, 3 months after the last dose of tocilizumab (if applicable), whichever is longer For male participants: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agree to refrain from donating sperm during the treatment period and for at least 3 months after pre-treatment with obinutuzumab or 2 months after the last dose of glofitamab, 90 days after the last dose of CC-220 and CC- 99282, 2 months after the final dose of tocilizumab (if applicable), whichever is longer  Exclusion Criteria:  Pregnancy or breastfeeding, or intention of becoming pregnant during the study or within 3 months after the final dose of mosunetuzumab, at least 3 months after pre-treatment with obinutuzumab or 2 months after the last dose of glofitamab, whichever is longer, 90 days after the last dose of CC-220 and CC-99282, 3 months after the final dose of tocilizumab (if applicable), whichever is longer Participant has received prior therapy with cereblon (CRBN)-modulating drug (e.g., lenalidomide, avadomide/CC-122, pomalidomide) </= 4 weeks prior to starting CC-220 and/or CC-99282 The following treatments prior to study entry: mosunetuzumab, glofitamab, or other CD20/CD3-directed bispecific antibodies; allogenic stem cell therapy (SCT); solid organ transplantation Treatments (investigation or approved) within the following time periods prior to initiation/first dose of study treatment: radiotherapy within 2 weeks; autologous SCT within 100 days; chimeric antigen receptor (CAR) T-cell therapy within 30 days; use of monoclonal antibodies or antibody-drug conjugates within 4 weeks; use of radioimmunoconjugates within 12 weeks; systemic immunosuppressive medications within 2 weeks; any other anti-cancer therapy, whether investigational or approved, including but not limited to chemotherapy, within 4 weeks or 5 half-lives of the drug, whichever is shorter Live, attenuated vaccine within 4 weeks before first dose of study treatment, or in whom it is anticipated that such a live attenuated vaccine will be required during the study period or within 5 months after the final dose of study treatment Current or past history of central nervous system (CNS) lymphoma or leptomeningeal infiltration History of severe allergic or anaphylactic reactions to humanized or murine monoclonal antibody therapy (or recombinant antibody-related fusion proteins) 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, vasculitis, or glomerulonephritis Major surgery or significant traumatic injury < 28 days prior to enrollment (excluding biopsies) or anticipation of the need for major surgery during study treatment Clinically significant toxicities from prior treatment have not resolved to Grade </= 1 (per US national cancer institute (NCI) common terminology criteria for adverse events (CTCAE) v5.0) prior to the first study drug administration with exceptions defined by the protocol Evidence of any significant, concomitant disease that could affect compliance with the protocol or interpretation of results For participants enrolled into glofitamab cohort: documented refractoriness to an obinutuzumab monotherapy-containing regimen