• 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