Filanesib and Carfilzomib in Treating Patients With Relapsed or Refractory Multiple Myeloma or Plasma Cell Leukemia

Inclusion Criteria: * Confirmed relapsed or refractory multiple myeloma (MM) or plasma cell leukemia (PCL); patients should have received at least 1 prior treatment regimen; prior treatment must have included at least one full cycle of a proteasome inhibitor (e.g., bortezomib) and at least one full cycle of an immunomodulatory (IMiD) (e.g., thalidomide, lenalidomide or pomalidomide); patients who have had prior ARRY-520 and carfilzomib will be allowed in the dose escalation phase, however prior ARRY-520 and carfilzomib will be excluded in the dose expansion cohort 1 of part A; there will be 2 cohorts in the dose expansion of part A: cohort 1 will be patients who are carfilzomib sensitive; cohort 2 will be patients who are carfilzomib refractory * Part B: For Part B dose-expansion: once a MTD has been established in part A, additional dose escalation will occur with subsequent dose escalation of carfilzomib; during the dose escalation of part B, patient (pt) must have at least 1 line of prior therapy and no limitations on prior therapy; patients who had prior clinical benefit/response to ARRY-520 or carfilzomib with a stable disease (SD) or better may be eligible for dose expansion of part B; dose expansion of part B will be patients who are carfilzomib sensitive * Measurable MM disease, defined as one of the following: * A monoclonal immunoglobulin (Ig) concentration on serum electrophoresis of \>= 0.5 g/dL for an IgG myeloma, \>= 0.1 g/dL for an IgD myeloma or 0.5 g/dL for an IgA myeloma * Measurable urinary light chain secretion by quantitative analysis of \>= 200 mg/24 hours * Involved serum free light chain (FLC) level \>= 10 mg/dL, provided the serum FLC ratio is abnormal * Patients with oligo- or non-secretory disease must have bone marrow involvement with at least 30% plasmacytosis * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0, 1 or 2 * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Platelets \>= 75 x 10\^9/L; if the bone marrow contains \>= 50% plasma cells, a platelet count of \>= 50 x 10\^9/L is allowed * Left ventricular ejection fraction (LVEF) \>= 40%; 2-dimensional (D) transthoracic echocardiogram (ECHO) is the preferred method of evaluation; multigated acquisition scan (MUGA) is acceptable if ECHO is not available * Aspartate aminotransferase (AST)/serum glutamic oxaloacetic transaminase (SGOT) and/or alanine aminotransferase (ALT)/serum glutamate pyruvic transaminase (SGPT) =\< 2.5 x the upper limit of normal (ULN) * Bilirubin \< 2.0 mg/dL * Serum creatinine =\< 2.5 mg/dL or a calculated creatinine clearance of at least 50 mL/min (using the Cockcroft and Gault method) * Female patients who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, (those who are post-menopausal for less than 1 year) must have negative serum or urine pregnancy test and agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent through 30 days after the last dose of study drug, or agree to completely abstain from heterosexual intercourse * Male patients, even if surgically sterilized (i.e., status postvasectomy), who: * Agree to practice effective barrier contraception during the entire study treatment period and through 30 days after the last dose of study drug, OR * Agree to completely abstain from heterosexual intercourse * Understand and voluntarily signed informed consent Exclusion Criteria: * Primary amyloidosis * Treatment with an investigational product or device within 21 days of cycle 1 day 1 * History of allergic reaction/hypersensitivity to any of the study medications, their analogues or excipients in the various formulations * Cytotoxic therapy or monoclonal antibodies within 21 days prior to cycle 1 day 1 * Radiotherapy within 21 days prior to cycle 1 day 1; however, if the radiation portal covered =\< 5% of the bone marrow reserve, the patient may be enrolled irrespective of the end date of radiotherapy * Major surgery within 14 days and minor surgery within 7 days prior to cycle 1 day 1 * Corticosteroid doses \> 10 mg/day of prednisone or equivalent within 14 days prior to cycle 1 day 1 * Medical, psychiatric, cognitive or other conditions that compromise the patient's ability to understand the patient information, to give informed consent, to comply with the study protocol or to complete the study or, in the judgment of the investigator, would make the patient inappropriate for study participation * Known history of allergy to Captisol (a cyclodextrin derivative used to solubilize carfilzomib) * Contraindication to any of the required concomitant drugs or supportive treatments, including hypersensitivity to all anticoagulation and antiplatelet options, antiviral drugs, or intolerance to hydration due to preexisting pulmonary or cardiac impairment * Patients who are eligible for autologous transplantation * Active congestive heart failure (New York Heart Association \[NYHA\] class III to IV), symptomatic ischemia, or conduction abnormalities uncontrolled by conventional intervention * Myocardial infarction within four months prior to enrollment * Lactating women * Patients with known human immunodeficiency virus (HIV) seropositivity * Patients with active clinical infections