Loncastuximab Tesirine and Mosunetuzumab for the Treatment of Relapsed or Refractory Diffuse Large B-Cell Lymphoma

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative. * Assent, when appropriate, will be obtained per institutional guidelines. * Be willing to provide tissue from a fresh core or excisional biopsy (performed as standard of care) of a tumor lesion prior to starting study therapy or from diagnostic tumor biopsies. * If unavailable, exceptions may be granted with study principal investigator (PI) approval. * Age: \>= 18 years. * Eastern Cooperative Oncology Group (ECOG) =\< 2. * Histologically confirmed diagnosis of diffuse large B-cell lymphoma or Follicular Lymphoma Grade 3B according to the World Health Organization (WHO) classification, with hematopathology review at the participating institution. Subtypes of DLBCL including transformed indolent lymphomas (TIL) including Richter's Transformation, primary mediastinal large B-cell lymphoma (PMBCL), and high-grade B-cell lymphoma not otherwise specified (HGBCL-NOS) are eligible. * Life expectancy \> 12 months. * Histologically confirmed diagnosis of diffuse large B-cell lymphoma (DLBCL) or follicular lymphoma Grade 3B according to the WHO classification, with hematopathology review at the participating institution. Subtypes of DLBCL including transformed indolent lymphomas (TIL) including Richter's Transformation, primary mediastinal large B-cell lymphoma (PMBCL), and high-grade B-cell lymphoma not otherwise specified (HGBCL-NOS) are eligible. * Relapsed or refractory disease after \>= 1 prior line of therapy (prior CD19-directed therapy and prior autologous stem cell transplant are allowed). * Relapse at the time of study enrollment must have been confirmed histologically (with hematopathology review at the participating institution). Exceptions may be granted with study PI approval. * Measurable disease by computerized tomography (CT) or positron emission tomography (PET)/CT scan with one or more sites of disease \>= 1.5 cm in longest dimension. * Tumor must be positive for both CD19 and CD20 by immunohistochemistry after the most recent therapy. * Fully recovered from the acute toxic effects (except alopecia) to =\< Grade 1 to prior anti-cancer therapy * Without bone marrow involvement: Absolute neutrophil count (ANC) \>= 1,000/mm\^3. * (G-CSF is allowed to reach ANC requirement). * With bone marrow involvement: no minimum ANC requirement. * (G-CSF is allowed to reach ANC requirement). * Platelets \>= 75,000/mm\^3. * Total bilirubin =\< 1.5 X upper limit of normal (ULN). * If hepatic involvement by lymphoma, or Gilbert's disease: =\< 3X ULN. * Aspartate aminotransferase (AST) =\< 2.5 x ULN. * If hepatic involvement by lymphoma: AST =\< 5 x ULN. * Alanine aminotransferase (ALT) =\< 2.5 x ULN. * If hepatic involvement by lymphoma: ALT =\< 5 x ULN . * Creatinine clearance of \>= 40 mL/min per 24 hour urine test or the Cockcroft-Gault formula * If not receiving anticoagulants: International Normalized Ratio (INR) OR Prothrombin (PT) =\< 1.5 x ULN. * If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants. * If not receiving anticoagulants: Activated Partial Thromboplastin Time (aPTT) =\< 1.5 x ULN * If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants. * Women of childbearing potential (WOCBP): negative urine or serum pregnancy test * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. * Agreement by females of childbearing potential to abstain from heterosexual intercourse or use two adequate method of birth control, including at least 1 method with a failure rate of \< 1% per year, for at least 28 days prior to Day 1 of Cycle 1, during the treatment period (including periods of treatment interruption), until 3 months after the final dose mosunetuzumab and 9 months after the last dose of loncastuximab tesirine. Women must refrain from donating eggs during this same period. Agreement by males to abstain from heterosexual intercourse or use a condom with female partners of childbearing potential or pregnant female partners during the treatment period and for 3 months after the final dose of mosunetuzumab and 6 months after the last dose of loncastuximab tesirine. Men must refrain from donating sperm during this same period. * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) with no identified cause other than menopause. * Examples of non-hormonal contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, established and proper use of progestogen only hormonal contraceptives that inhibit ovulation, hormone releasing intrauterine devices, and copper intrauterine devices. Barrier methods must always be supplemented with the use of a spermicide. Note: Combined oral contraceptives are not recommended. Exclusion Criteria: * Prior treatment with loncastuximab tesirine. * Prior treatment with mosunetuzumab or other CD20-directed bispecific antibodies. * Prior allogeneic stem cell transplantation. * Prior use of any monoclonal antibody, radioimmunoconjugate or ADC within 2 weeks prior to Day 1 of protocol therapy. * Treatment with any chemotherapeutic agent, or treatment with any other anti-cancer agent (investigational or otherwise) within 2 weeks or 5 half-lives of the drug, whichever is shorter, prior to Day 1 of protocol therapy. * Treatment with radiotherapy within 2 weeks prior to Day 1 of protocol therapy. * If patients have received radiotherapy within 4 weeks prior to prior to Day 1 of protocol therapy, patients must have at least one measurable lesion outside of the radiation field. Patients who have only one measurable lesion that was previously irradiated but subsequently progressed are eligible. * Autologous stem cell transplantation (SCT) within 30 days prior to prior to Day 1 of protocol therapy. * Prior treatment with chimeric antigen receptor T-cell (CAR-T) therapy within 30 days prior to Day 1 of protocol therapy. * Live vaccine within 30 days prior to Day 1 of protocol therapy. * Concomitant investigational therapy. * Treatment-emergent immune-related adverse events associated with prior immunotherapeutic agents(e.g., immune checkpoint inhibitor therapies). Note: For certain prior treatments, such as CAR-T cell therapies, patients with prior immune-related Grade \>= 3 adverse events (e.g., CRS) may be allowed after discussion with and approval by the Study PI. * Systemic steroid therapy or any other form of immunosuppressive therapy for lymphoma symptom control must be tapered down to =\< 20 mg/day prednisone or equivalent. Exceptions are: * Inhaled or topical steroids * Use of mineralocorticoids for management of orthostatic hypotension * Use of physiologic doses of corticosteroids for management of adrenal insufficiency * Known hypersensitivity to biopharmaceutical produced in chinese hamster ovary (CHO) cells or history of allergic reactions attributed to compounds of similar chemical or biologic composition to study agents. * 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). * History of solid organ transplantation. * History of progressive multifocal leukoencephalopathy (PML). * Known or suspected history of hemophagocytic lymphohistiocytosis (HLH). * Clinically significant uncontrolled illness. * Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment. * Known active hepatitis B virus (HBV) or hepatitis C virus (HCV) infection. Patients with past HBV infection (defined as negative hepatitis B surface antigen \[HBsAg\] and positive hepatitis B core antibody \[HBcAb\]) are eligible if HBV deoxyribonucleic acid (DNA) is undetectable. Patients who are positive for HCV antibody are eligible if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA). Testing to be done only in patients suspected of having infections or exposures. * Known active human immunodeficiency virus (HIV) infection. Subjects who have an undetectable or unquantifiable HIV viral load with CD4 \> 200 and are on HAART medication are allowed. Testing to be done only in patients suspected of having infections or exposures. * Known or suspected chronic active Epstein-Barr virus (EBV) infection. * Known active central nervous system (CNS) involvement by lymphoma, including leptomeningeal involvement. * History of erythrema multiforme, Grade \>= 3 rash, or blistering following prior treatment with immunomodulatory derivatives. * Symptomatic cardiac disease (including symptomatic ventricular dysfunction, symptomatic coronary artery disease, and symptomatic arrhythmias), cerebrovascular event/stroke or myocardial infarction within the past 6 months. * Clinically significant history of liver disease, including viral or other hepatitis, or cirrhosis. * Active autoimmune disease requiring treatment. * History of autoimmune disease, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener granulomatosis, Sjogren syndrome, Guillain-Barre syndrome, multiple sclerosis, vasculitis, or glomerulonephritis. * Patients with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone may be eligible. * Patients with controlled Type 1 diabetes mellitus who are on an insulin regimen are eligible for the study. * Patients with a history of disease-related immune thrombocytopenic purpura, autoimmune hemolytic anemia, or other stable autoimmune diseases may be eligible. * Recent major surgery (within 4 weeks) prior to start of protocol therapy, other than for diagnosis. * History of another primary malignancy that has not been in remission for at least 2 years, with the following exceptions: * Adequately treated non-melanoma skin cancer or lentigo maligna (melanoma in situ) without evidence of disease * Adequately treated in situ carcinomas (e.g. cervical, esophageal) without evidence of disease * Asymptomatic prostate cancer managed with a watch-and-wait strategy * If the malignancy is expected to not require any treatment for at least 2 years (this exception should be discussed with the study PI). * Females only: Pregnant or breastfeeding. * Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures. * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics).