Daratumumab, Bortezomib, and Dexamethasone Followed by Daratumumab, Ixazomib, and Dexamethasone in Treating Patients With Relapsed or Refractory Multiple Myeloma

Inclusion Criteria: * Participants must have histologically confirmed diagnosis of multiple myeloma * Participants must have measurable disease, as defined by at least one of the following: * Serum monoclonal protein M-protein level \>= 0.5 g/dL * Urinary M-protein excretion of \>= 200 mg over a 24-hour period * Involved free light chain level \>= 10 mg/dL, along with an abnormal free light chain ratio * Participants must have disease that has relapsed and/or refractory after their most recent therapy, with progressive disease (PD) being defined as an increase of 25% from the lowest response value in any one or more of the following: * Serum M-component protein (the absolute increase must be \>= 0.5 g/dL) and/or * Urine M-component protein (the absolute increase must be \>= 200 mg/24 hours) and/or * Only in participants without a measurable serum and urine M protein level: the difference between involved and uninvolved free light chain (FLC) levels (absolute increase) must be \> 10 mg/dL * Definite development of new bone lesions or soft tissue plasmacytomas or definite increase in the size of existing bone lesions or soft tissue plasmacytomas * Development of hypercalcemia (corrected serum calcium \> 11.5 mg/dL) that can be attributed solely to the plasma cell proliferative disorder * Participants with one to three lines of therapy for their disease with a prior therapy is defined as 2 or more cycles of therapy given as a treatment plan for multiple myeloma (MM) (e.g. a single-agent or combination therapy or a sequence of planned treatments such as induction therapy followed by autologous stem cell transplant (SCT) and then consolidation and/or maintenance therapy) * Participants must have achieved a partial response or better to at least one prior line of therapy * Participants must have an Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status 0, 1, or 2 * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 without growth factors within 2 weeks of initiation of treatment * Platelets \>= 75,000/mm\^3 * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN). In subjects with documents Gilbert's syndrome, total bilirubin =\< 2 x ULN * Aspartate aminotransferase (AST) serum glutamic oxaloacetic transaminase (SGOT)/alkaline phosphatase (ALT) serum glutamic pyruvic transaminase (SGPT) =\< 3 x ULN * Creatinine clearance \>= 30 mL/min/1.73 m\^2 * Participants must be willing to give written consent before performance of any study related procedures not part of standard medical care, with the understanding that consent may be withdrawn by the participant at any time without prejudice to future medical care * Female participants who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent form through 90 days after the last dose of study drug, OR * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the participant. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception) * Male participants, even if surgically sterilized (i.e., status post-vasectomy), must agree to one of the following: * Agree to practice effective barrier contraception during the entire study treatment period and through 90 days after the last dose of study drug, OR * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the participants. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception) Exclusion Criteria: * Participants who received prior ixazomib at any time or daratumumab or other anti-CD38 therapies, except as part of initial therapy if this was stopped to move on to SCT and the participant did not progress on anti-CD38 treatment * Participants are refractory to bortezomib or carfilzomib at the last exposure before this study (defined as subject having PD while receiving bortezomib or carfilzomib therapy or within 60 days after ending bortezomib or carfilzomib therapy) * Participants with known allergy to any of the study medications or their analogues * Participants planning to undergo SCT prior to PD on this study (i.e., these subjects should not be enrolled in order to reduce disease burden prior to transplant) * Participants receiving systemic treatment with strong cytochrome P450, family 3, subfamily A (CYP3A) inducers (e.g. rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital) or use of St. John's wort within 14 days before randomization * Participants must have completed their most recent drug therapy directed at multiple myeloma in the following timeframes: * Antitumor therapy (chemotherapy, molecular targeted therapy, retinoid therapy, or hormonal therapy,) within 14 days of study day 1. * Antibody therapy within 28 days of study day 1 * Investigational drug (including investigational vaccine) or invasive investigational medical device for any indication within 28 days or 5 pharmacokinetic half-lives, whichever is longer, of study day 1. * Corticosteroids at least 14 days prior to starting therapy, except for a dose equivalent to dexamethasone of =\< 4 mg/day OR an emergency use of a short course of corticosteroids (equivalent of dexamethasone 40 mg/day for a maximum of 4 days) * Autologous stem cell transplantation at least 12 weeks prior to starting study treatment * Allogeneic stem cell transplantation at least 24 weeks prior to starting treatment, and these Participants must also NOT have moderate to severe active acute or chronic graft versus host disease (GVHD) * Participants with known chronic obstructive pulmonary disease (COPD) with a forced expiratory volume in 1 second (FEV1) \<50% of predicted normal. * Participants with Grade 2 or higher residual toxicities from prior therapy (including Grade 2 or higher peripheral neuropathy or any grade neuropathy with pain) with the exception of alopecia. * Participants who have undergone major surgery within 28 days of study Day 1. NOTE: Subjects with planned surgical procedures to be conducted under local anesthesia may participate. Kyphoplasty or vertebroplasty are not considered major surgery. * Participants with central nervous system involvement of myeloma. * Participants with uncontrolled intercurrent illness including, but not limited to, ongoing or active infection requiring intravenous antibiotics or psychiatric illness/social situations that would limit compliance with study requirements. * Participants with myocardial infarction within 6 months of study day 1, symptomatic congestive heart failure (New York Heart Association (NYHA) Class III and higher), unstable angina, or uncontrolled cardiac arrhythmia (Grade 2 or higher). * Participants with GI tract disease causing the inability to take oral medication, malabsorption syndrome, requirement for intravenous alimentation, prior surgical procedures affecting absorption, uncontrolled inflammatory GI disease (e.g., Crohn's disease, ulcerative colitis). * Participants with active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive. * Participants diagnosed or treated for another malignancy within 2 years before study enrollment or previously diagnosed with another malignancy and have any evidence of residual disease. Subjects with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection * Female participants who are lactating or have a positive serum pregnancy test during the screening period. * Radiotherapy within 14 days before enrollment. If the involved field is small, 7 days will be considered a sufficient interval between treatment and administration of the ixazomib. * Participation in other therapeutic clinical trials, including those with other investigational agents not included in this trial, within 30 days of the start of this trial. * Participants that have previously participated in a study with ixazomib whether treated with ixazomib or not.