Inclusion Criteria:  Pathologic diagnosis of DLBCL, as defined by the 2016 World Health Organization (WHO) classification (including participants with DLBCL transformed from indolent lymphoma), or high-grade B cell lymphoma (HGBCL), or Grade 3b follicular lymphoma (FL). Measurable disease as defined by the 2014 Lugano Classification. Stages I-IV. ECOG PS 0-2; ECOG PS 3 allowed if status is deemed related to lymphoma & felt to be potentially reversible by the treating physician.  Adequate organ function as defined by screening laboratory values within the following parameters:  Absolute neutrophil count (ANC) ≥1.0 x 10^3/µL (off growth factors at least 72 hours). Platelet count ≥75 x 10^3/µL without transfusion in the past 7 days. Alanine aminotransferase (ALT), aspartate aminotransferase (AST), and gamma glutamyl transferase (GGT) ≤2.5 x the upper limit of normal (ULN). Total bilirubin ≤1.5 x ULN (participants with known Gilbert's syndrome may have a total bilirubin up to ≤3 x ULN). Calculated creatinine clearance >30 mL/min by the Cockcroft and Gault equation.  Note: A laboratory assessment may be repeated a maximum of two times during the screening period to confirm eligibility.  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 12 months after the last dose of study treatment. 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 until at least 6 months after the participant receives his last dose of study treatment.  Inclusion Criteria specific for Cohort A:  Unfit as defined by the simplified geriatric assessment (sGA). Includes all of the following:  Aged ≥80 years ADL score of 6 IADL score of 8 CIRS-G: no score of 3-4 and <5 scores of 2  Inclusion Criteria specific for Cohort B:  Frail as defined by sGA (Includes all of the following):  Aged ≥80 years ADL score of <6 IADL score of <8 CIRS-G: ≥1 score of 3-4 and >5 scores of 2 OR  Aged ≥65 with at least one of the following cardiac comorbidities that make anthracycline-containing regimens inadvisable as determined by the investigator.  Left ventricular ejection fraction (LVEF) ≥30 to <50% History of myocardial infarction within 6 months prior to screening Ischemic heart disease History of stroke within 12 months prior to screening  Exclusion Criteria:  Known history of hypersensitivity to or positive serum human anti-drug antibody to a cluster of differentiation 19 (CD19) antibody. Previous therapy for DLBCL, HGBCL, or Grade 3b FL (with exception of corticosteroid course for symptom management of less than 14 days). Previous therapy with loncastuximab tesirine or R-CHOP for any indication. Low-risk (score 0-1) per Elderly Prognostic Index (EPI) . Known history of hypersensitivity to any component of study treatment (loncastuximab tesirine, rituximab and/or R-CHOP respectively).  Human immunodeficiency virus (HIV) seropositive with any of the following:  CD4+ T-cell (CD4+) counts <350 cells/µL Acquired immunodeficiency syndrome (AIDS) - defining opportunistic infection within 12 months prior to screening Not on anti-retroviral therapy, or on anti-retroviral therapy for <4 weeks at the time of screening HIV viral load ≥400 copies/mL 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 Stevens-Johnson syndrome or toxic epidermal necrolysis. Lymphoma with active central nervous system involvement at the time of screening, including leptomeningeal disease. 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). Major surgery, radiotherapy, chemotherapy, or other anti-neoplastic therapy within 14 days prior to start of study drug (Cycle 1 Day 1 [C1D1]), except shorter if approved by the Sponsor. Use of any other experimental medication within 14 days prior to start of study drug (C1D1). Received live vaccine within 4 weeks of C1D1. Congenital long QT syndrome or a corrected Fridericia correction of the QT measure (QTcF) interval of >480 ms at screening (unless secondary to pacemaker or bundle branch block). 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. Any other significant medical illness, abnormality, or condition that would, in the Investigator's judgment, make the participant inappropriate for study participation or put the participant at risk.