A Study to Evaluate the Safety and Anti-cancer Activity of Loncastuximab Tesirine in Combination With Other Anti-cancer Agents in Participants With Relapsed or Refractory B-cell Non-Hodgkin Lymphoma (LOTIS-7)

Inclusion Criteria: * Male or female participant aged 18 years or older * Pathologic diagnosis of relapsed (disease that has recurred following a response) or refractory (disease that failed to respond to prior therapy) B-NHL (2016 World Health Organization classification) who have failed, or been intolerant to any approved therapy and had received at least two systemic treatment regimens in Part 1; and at least one systemic treatment regimen in Part 2 * LBCL: Part 2 Arm E enrollment focused on LBCL only * DLBCL, not otherwise specified (NOS) * Germinal Center B-cell type * Activated B-cell type * Transformed FL (note: patients with transformed FL must have received at least one line of systemic therapy post-transformation to be eligible) * HGBCL, with MYC and BCL2 and/or BCL6 rearrangements * HGBCL, NOS * FL Grade 3b * Arm F and Part 1 Arm E: * All LBCL histologies listed above * FL (Grade 1-3a) * MZL * For Arm C only: * All histologies listed above * DLBCL (including transformed diseases) * MCL * BL * Life expectancy of at least 24 weeks according to Investigator's judgement * Need of systemic treatment for any of the listed indications as assessed by the investigator, including indolent B-NHLs (e.g. FL and MZL) * Measurable disease as defined by the 2014 Lugano Classification * Availability of formalin-fixed paraffin-embedded tumor tissue block * ECOG performance status 0 to 2 * Adequate organ function * Women of childbearing potential (WOCBP) must agree to use a highly effective method of contraception from the time of giving informed consent until at least 10 months after the last dose of loncastuximab tesirine. Men with female partners who are of childbearing potential must agree to use a condom when sexually active or practice total abstinence from the time of giving informed consent the first dose until at least 7 months after the last dose of loncastuximab tesirine. Men must refrain from donating sperm during this same period. Arm E: WOCBP must agree to use contraceptive methods that result in a failure of less than 1% per year or remain abstinent (refrain from heterosexual intercourse) during the treatment period and for at least 18 months after pretreatment with obinutuzumab. Arm F: WOCBP must agree to use contraceptive methods that result in a failure of less than 1% per year or remain abstinent (refrain from heterosexual intercourse) during the treatment period and for at least 3 months after the final dose of mosunetuzumab and tocilizumab (if applicable) * Patients 80 years of age and older at the time of signing the informed consent must be deemed fit by Cumulative Illness Rating Scale - Geriatric (CIRS-G scale), defined as no score of 3-4 in any category AND \< 5 categories with a score of 2 excluding hematologic criteria Exclusion Criteria: * Known history of hypersensitivity resulting in treatment discontinuation to or positive serum human ADA to a CD19 antibody * Previous therapy with loncastuximab tesirine * Previous treatment with polatuzumab vedotin, glofitamab or mosunetuzumab (applied to relevant arm and/or cohort of the specific drug administered) * Participants who received previous treatment of polatuzumab vedotin containing regimen will be excluded from Arm C * Participants who received previous treatment of glofitamab containing regimen will be excluded from Arm E * Participants who received previous treatment of mosunetuzumab containing regimen will be excluded from Arm F * Human immunodeficiency virus (HIV) seropositive * Serologic evidence of chronic hepatitis B virus (HBV) infection and unable or unwilling to receive standard prophylactic antiviral therapy or with detectable HBV viral load * Serologic evidence of hepatitis C virus (HCV) infection without completion of curative treatment or with detectable HCV viral load * History of confirmed progressive multifocal leukoencephalopathy * History of Stevens-Johnson syndrome, toxic epidermal necrolysis, or macrophage activation syndrome (MAS)/hemophagocytic lymphohistiocytosis (HLH)/immune effector cell associated HLH-like syndrome (IEC-HS) * Existing pericardial effusion (any grade) or clinically significant third space fluid accumulation (i.e., ascites requiring drainage or pleural effusion that is either requiring drainage or associated with shortness of breath) * Breastfeeding or pregnant * Significant medical comorbidities * Major surgery, radiotherapy, chemotherapy, or other anti-neoplastic therapy, within 14 days prior to start of study drugs (C1 D1), unless approved by the Sponsor * Live vaccine within 4 weeks prior to C1D1 * Failure to recover to Grade ≤1 (Common Terminology Criteria for Adverse Events \[CTCAE\] version 5.0) from acute non-hematologic toxicity (excluding alopecia) due to previous therapy prior to screening * Active second primary malignancy other than non-melanoma skin cancers, non-metastatic prostate cancer, in situ cervical cancer, ductal or lobular carcinoma in situ of the breast, or other malignancy that the Sponsor's medical monitor and Investigator agree and document should not be exclusionary Extra Exclusion Criteria for Arms E (includes glofitamab) and F (includes mosunetuzumab) Note: as applicable, the arm-specific exclusion criteria may supersede the general ones, such as stem cell transplant. * Prior allogeneic stem cell transplant and solid organ transplant * Autologous stem cell transplant within 100 days prior to C1D1 * History of central nervous system (CNS) lymphoma or leptomeningeal infiltration * Current or history of CNS disease, such as stroke, epilepsy, CNS vasculitis, or neurodegenerative disease * Known active infection, reactivation of a latent infection, whether bacterial, viral, fungal, mycobacterial, or other pathogens (excluding fungal infections of nail beds), or any major episode of infection requiring hospitalization or treatment with intravenous (IV) antibiotics within four weeks prior to C1D1 * Active or history of autoimmune disease or immune deficiency, motor neuropathy considered of autoimmune origin and other CNS autoimmune diseases, including but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjögren syndrome, Guillain-Barré syndrome, or multiple sclerosis, with, with certain exceptions * Prior treatment with anti-cancer/lymphoma targeted therapies (e.g., tyrosine kinase inhibitors, systemic immunotherapeutic/immunostimulating agents, including, but not limited to, cluster of differentiation 137 agonists or immune checkpoint blockade therapies, including anti-cytotoxic T-lymphocyte-associated antigen 4 (CTLA-4), anti-programmed cell death protein 1 (PD1), and anti-programmed death ligand 1 (PDL1) therapeutic antibodies, radio-immunoconjugates, ADCs, immune/cytokines and monoclonal antibodies) or treatment with systemic immunosuppressive medication (including, but not limited to, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor agents) within 4 weeks or five half-lives of the drug, whichever is shorter, prior to C1D1, or anticipation of need for systemic immunosuppressive medication during study treatment, with certain exceptions * Prior treatment with CAR-T-cell therapy within 100 days prior to C1D1; primary refractory patients (progressive or persistent disease within 30 days) to CAR-T-cell therapy are not eligible * Toxicities from prior anti-cancer therapy including immunotherapy that did not resolve to ≤ Grade 1 with the exception of alopecia, endocrinopathy managed with replacement therapy and stable vitiligo * Any history of immune-related Grade ≥3 AE with the exception of endocrinopathy managed with replacement therapy * Ongoing corticosteroid use greater than 25 mg/day of prednisone or equivalent within 4 weeks prior and during study treatment * Administration of a live attenuated vaccine within 4 weeks prior to the first dose of study treatment or anticipation that such a live attenuated vaccine will be required during the study or within 5 months after last dose of study treatment * Arm E only: Known history of hypersensitivity to obinutuzumab