Ixazomib Citrate in Treating Patients With Relapsed Multiple Myeloma That Is Not Refractory to Bortezomib

Inclusion Criteria:

* Calculated creatinine clearance (using Cockcroft-Gault equation) \>= 30 mL/min (obtained =\< 14 days prior to registration)
* Absolute neutrophil count \>= 1000/mL (obtained =\< 14 days prior to registration)
* Untransfused platelet count \>= 75000/mL (obtained =\< 14 days prior to registration)
* Hemoglobin \>= 8.0 g/dL (obtained =\< 14 days prior to registration)
* Total bilirubin =\< 1.5 x the upper limit of the normal range (ULN) (obtained =\< 14 days prior to registration)
* Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN (obtained =\< 14 days prior to registration)
* Patients with relapsed multiple myeloma who have already received one or more standard treatment regimens
* Measurable disease of multiple myeloma as defined by at least ONE of the following:

  * Serum monoclonal protein \>= 1.0 g/dL
  * \>= 200 mg of monoclonal protein in the urine on 24 hour electrophoresis
  * Serum immunoglobulin free light chain \>= 10 mg/dL AND abnormal serum immunoglobulin kappa to lambda free light chain ratio
  * For patients with extramedullary disease (EMD) measurable disease by computed tomography (CT) or magnetic resonance imaging (MRI) or the CT portion of the positron emission tomography (PET)/CT: Must have at least one lesion that has a single diameter of \>= 2 cm. Skin lesions can be used if the area is \>= 2 cm in at least one diameter and measured with a ruler
  * Plasma cell count \>= 0.5 x 10\^9/L or 5 percent of the peripheral blood white cells
  * Plasma cell count if determined by flow cytometry, \>= 200/150,000 events
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, 2
* Provide informed written consent
* Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only
* Willing to return to consenting Mayo Clinic institution for follow-up during the Active Monitoring Phase of the study; Note: during the Active Monitoring Phase of a study (i.e., active treatment and observation), participants must be willing to return to the consenting institution for follow-up
* Recovered (i.e., \< grade 1 toxicity) from the reversible effects of prior antineoplastic therapy
* Arms A - D only: Patients should be proteasome inhibitor naive (including bortezomib and carfilzomib) OR have received less than 6 cycles of therapy with a bortezomib or carfilzomib containing regimen and were not refractory to the bortezomib or carfilzomib based regimen (less than a PR or progression on or within 60 days of discontinuation)
* Arm E only: Negative hepatitis B test (defined by a negative test for hepatitis B surface antigen \[HBsAg\], or antibodies to hepatitis B surface and/or core antigens \[antiHBs or antiHBc\]) (added as of addendum 9); Note: patients with serologic findings suggestive of hepatitis B virus (HBV) vaccination (antiHBs positivity as the only serologic marker) AND a known history of prior HBV vaccination do not need to be tested for HBV deoxyribonucleic acid (DNA) by polymerase chain reaction (PCR); those who are PCR positive will be excluded

Exclusion Criteria:

* Recent prior chemotherapy:

  * Alkylators (e.g. melphalan, cyclophosphamide) =\< 14 days prior to registration
  * Anthracyclines =\< 14 days prior to registration
  * High dose corticosteroids, immune modulatory drugs (thalidomide or lenalidomide) =\< 7 days prior to registration
* Prior therapy with any proteasome inhibitor other than bortezomib, carfilzomib, or ixazomib
* Concomitant high dose corticosteroids other than what is part of treatment protocol (concurrent use of corticosteroids); EXCEPTION: patients may be on chronic steroids (maximum dose 20 mg/day prednisone equivalent) if they are being given for disorders other than myeloma, i.e., adrenal insufficiency, rheumatoid arthritis, etc
* 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; patients with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection
* Any of the following:

  * Pregnant women or women of reproductive ability who are unwilling to use 2 effective methods of contraception from the time of signing the informed consent form through 90 days after the last dose of study drug
  * Nursing women
  * Men who are unwilling to use a condom (even if they have undergone a prior vasectomy) while having intercourse with any woman, while taking the drug and for 30 days after stopping treatment
* Other co-morbidity which would interfere with patient's ability to participate in trial, e.g. uncontrolled infection, uncompensated heart or lung disease
* Other concurrent chemotherapy, radiotherapy, or any ancillary therapy considered investigational; NOTE: bisphosphonates are considered to be supportive care rather than therapy, and are thus allowed while on protocol treatment
* Patient has \>= grade 3 peripheral neuropathy, or grade 2 with pain on clinical examination during the screening period
* Major surgery within 14 days before study registration
* Systemic treatment with strong inhibitors of cytochrome P450, family 1, subfamily A (CYP3A) inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of Ginkgo biloba or St. John's wort within 14 days before the first dose of study treatment
* Evidence of current uncontrolled cardiovascular conditions, including cardiac arrhythmias, congestive heart failure, angina, or myocardial infarction within the past 6 months; Note: prior to study entry, any electrocardiogram (ECG) abnormality at screening must be documented by the investigator as not medically relevant
* Known human immunodeficiency virus (HIV) positive
* Known hepatitis B surface antigen-positive status, or known or suspected active hepatitis C infection
* Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol
* Known allergy to any of the study medications, their analogues or excipients in the various formulations
* Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib including difficulty swallowing
* Diarrhea \> grade 1, based on the National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) grading, in the absence of antidiarrheals
* Arm E only: Refractory to any combination of a proteasome inhibitor and daratumumab
* Arm E only: Known chronic obstructive pulmonary disease with a forced expiratory volume in 1 second (FEV1) \< 50% of predicted normal. (Note that FEV1 testing is required for subjects suspected of having chronic obstructive pulmonary disease and subjects must be excluded if FEV1 \< 50% of predicted normal.)
* Arm E only: Known moderate or severe persistent asthma within the past 2 years or currently has uncontrolled asthma of any classification