Inclusion Criteria:

* INCLUSION CRITERIA FOR SCREENING (STEP 0 - PREREGISTRATION)
* Age \>= 18 and =\< 70 years
* Patients must have histologically confirmed mantle cell lymphoma, with cyclin D1 by immunohistochemical stains and/or t(11;14) by cytogenetics or fluorescence in situ hybridization (FISH). If patient has cyclin D1 negative mantle cell lymphoma with classical morphology and an expression profile (including SOX11+) that is otherwise indistinguishable from mantle cell lymphoma, communication with investigator is required for consideration of enrollment. The proliferation rate, using Ki-67 or MIB-1, should also be determined, but is not required until step 1 registration; patients may register to step 0 without a documented Ki-67 index
* In the opinion of the enrolling physician, patients must be felt to be a candidate for autologous stem cell transplantation
* Patient may be about to begin, be receiving or have completed induction therapy within 120 days prior to preregistration to step 0; no more than 300 days may have passed between the first day of induction therapy and preregistration to step 0

* For patients who have completed induction therapy and have been restaged, restaging evaluation must show status of partial (PR) or complete response (CR); post-induction patients with evidence of clinical disease progression are not eligible for preregistration
* Up to two regimens of therapy (conventional chemotherapy, antibody therapy, or an oral regimen) are allowed as long as a continuous response was ongoing throughout therapy; overall, a partial response needs to have been achieved (using studies at the time of diagnosis as the baseline)

* NOTE: For example, a patient who started treatment with rituximab/bendamustine and was then switched to rituximab(R)-cyclophosphamide, doxorubicin hydrochloride, vincristine sulfate, and prednisone (CHOP) (due to insufficient response or excessive toxicity) would be counted as having received 2 regimens; however, R-CHOP alternating with R-dexamethasone, high-dose cytarabine, and cisplatin (DHAP) as a planned induction regimen would count as one regimen
* Patient does not have any documented history of central nervous system (CNS) involvement by mantle cell lymphoma; this includes no evidence of parenchymal brain, spinal cord, or cerebrospinal fluid involvement; radiculopathy symptoms from nerve root compression by lymphoma do not constitute CNS involvement
* Patient must have archived formalin-fixed paraffin-embedded (FFPE) tumor tissue specimen from the original diagnostic biopsy available for submission to Adaptive Biotechnologies for ClonoSEQ ID molecular marker identification of unique clonal immunoglobulin deoxyribonucleic acid (DNA) sequence

* NOTE: If adequate tumor tissue is not available, peripheral blood collected prior to start of treatment with high disease burden (\> 5%) is acceptable for molecular marker identification (ID) testing

* Adaptive Biotechnologies will forward results within fourteen (14) days of receipt of any stored (e.g. frozen or FFPE) tumor tissue specimen to the submitting institution and to the ECOG-American College of Radiology Imaging Network (ACRIN) Operations Office
* NOTE: Patients for whom the molecular marker is identified will have peripheral blood collected after completion of induction (patient's disease status is PR or CR) and submitted to Adaptive Biotechnologies for minimal residual disease (MRD) assessment

* Adaptive Biotechnologies will forward results within ten to fourteen (10-14) days of receipt of fresh peripheral blood specimen to the submitting institution and to the ECOG-ACRIN Operations Office
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Patients must have met eligibility criteria for the screening step 0
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): The proliferation rate, using Ki-67 or MIB-1 immunohistochemistry (=\< 30% versus \> 30% versus "indeterminate" Ki-67 index), must be documented for a baseline tumor biopsy specimen
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Institution has received results from Adaptive Biotechnologies as defined by one of the following criteria:

* Patients are "MRD Indeterminate": ClonoSEQ ID molecular marker assessment did not identify any unique clonal immunoglobulin DNA sequence OR
* ClonoSEQ ID molecular marker assessment identified unique clonal immunoglobulin DNA sequence and MRD assessment is completed
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Patients must have completed induction therapy within 150 days prior to registration to step 1, AND no more than 300 days may have elapsed from the first dose of induction chemotherapy (cycle 1 \[C1\] day 1 \[D1\]) given, until the last day of induction chemotherapy administered; for those assigned to Arms A, C, or D, the date of transplant ("day 0") must not be greater than 365 days after the first dose of induction chemotherapy (C1D1) given

* Patient must have received at least four (4) cycles of induction therapy
* Up to two regimens of therapy (conventional chemotherapy, antibody therapy, or an oral regimen) are allowed as long as a continuous response was ongoing throughout therapy

* NOTE: For example, a patient who started treatment with rituximab/bendamustine and was then switched to R-CHOP (due to insufficient response or excessive toxicity) would be counted as having received 2 regimens; however, R-CHOP alternating with R-DHAP as a planned induction regimen would count as one regimen
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Patients must have achieved a radiologic complete or partial remission as defined by the Lugano criteria
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): In the opinion of the enrolling physician, patients must be felt to be a candidate for autologous stem cell transplantation
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Patients have an Eastern Cooperative Oncology Group (ECOG) performance status of 0-2
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Human immunodeficiency virus (HIV) positive patients are not excluded, but to enroll, must meet all of the below criteria:

* HIV is sensitive to antiretroviral therapy
* Must be willing to take effective antiretroviral therapy that has minimal overlapping toxicity and pharmacokinetic interactions with protocol therapy
* No history of HIV-related opportunistic disease or acquired immune deficiency syndrome (AIDS)-defining conditions within past 12 months other than historic CD4+ T-cell counts below 200 cells/mm\^3
* Expected long-term survival if lymphoma were not present
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Patient must be disease-free \>= 3 years of prior malignancies with the exception of adequately treated non-melanoma skin cancer, adequately treated in situ carcinoma, melanoma in situ post wide local excision or Mohs surgery, low grade prostate carcinoma (Gleason grade =\< 6) managed with observation that has been stable for at least 6 months
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Patient must not be pregnant or breast-feeding due to the potential for congenital abnormalities and of harm to nursing infants due to the treatment regimens used

* All patients of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy
* A patient of childbearing potential is anyone, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months)
* INCLUSION CRITERIA FOR TREATMENT ASSIGNMENT (STEP 1): Patient of childbearing potential and sexually active males must be strongly advised to use an accepted and effective method of contraception or to abstain from sexual intercourse for the duration of their participation in the study and for 12 months post rituximab treatment