Brief Summary: This phase II trial studies the safety and how well of loncastuximab tesirine when given together with mosunetuzumab works in treating patients with diffuse large B-cell lymphoma that has come back (relapsed) or does not respond to treatment (refractory). Loncastuximab tesirine is a monoclonal antibody, loncastuximab, linked to a toxic agent called tesirine. Loncastuximab attaches to anti-CD19 cancer cells in a targeted way and delivers tesirine to kill them. Mosunetuzumab is a monoclonal antibody that may interfere with the ability of cancer cells to grow and spread. Giving loncastuximab tesirine with mosunetuzumab may help treat patients with relapsed or refractory diffuse large B-cell lymphoma.

Detailed Description: PRIMARY OBJECTIVES: I. Evaluate the safety and tolerability of loncastuximab tesirine plus mosunetuzumab in patients with relapsed or refractory (R/R) diffuse large B-cell lymphoma (DLBCL). (Safety Lead-In) II. Estimate the overall response rate (ORR) in R/R DLBCL patients treated with loncastuximab tesirine plus mosunetuzumab. (Phase 2) SECONDARY OBJECTIVES: I. Estimate the complete response (CR) rate, duration of response (DOR), progression-free survival (PFS), and overall survival (OS) in R/R DLBCL patients treated with loncastuximab tesirine plus mosunetuzumab. II. Evaluate the toxicity of loncastuximab tesirine plus mosunetuzumab for R/R DLBCL. EXPLORATROY OBJECTIVE: I. Evaluate genomic and immune biomarkers of response and resistance to loncastuximab tesirine combined with mosunetuzumab OUTLINE: Patients receive loncastuximab tesirine intravenously (IV) and mosunetuzumab IV on study. Patients also undergo positron emission tomography (PET)/computed tomography (CT) scan, biopsy, and collection of blood samples on study.