Nonmyeloablative Haploidentical Transplant Followed by MLN9708

Inclusion Criteria:

* Availability of a 3/6 - 5/6 matched (HLA-A, B, DR) related donor
* Donor must have negative HLA cross-match in the host vs. graft direction.
* Donor must be willing to donate mobilized peripheral blood stem cells
* Age ≥ 18 years
* Karnofsky status ≥ 70%
* One of the following high-risk malignancies:
* Chronic Myelogenous Leukemia (chronic phase, resistant and/or intolerant to tyrosine kinase inhibitors (OR) accelerated phase (OR) blast crisis in 2nd chronic phase following induction chemotherapy)
* Acute Myelogenous Leukemia (2nd or subsequent complete remission \[CR\] (OR) Primary induction chemotherapy failure, but subsequently entered into a CR(OR) 1st CR with poor risk cytogenetics or molecular markers; or arising from preceding hematological disease)
* Myelodysplastic Syndrome (treatment-related, monosomy 7 or complex cytogenetics, IPSS score of 1.5 or greater, Chronic myelomonocytic leukemia \[CMML\])
* Acute lymphocytic leukemia/lymphoblastic lymphoma (2nd or subsequent CR (OR) Primary induction chemotherapy failure, but subsequently entered into a CR (OR) 1st CR with poor risk cytogenetics)
* Chronic Lymphocytic Leukemia / Prolymphocytic Leukemia (Duration of remission \<12 months after receiving chemotherapy with a nucleoside analog (OR) High risk features (i.e. 17p deletion), (OR) Second or subsequent relapse)
* Hodgkin's or Non-Hodgkin's Lymphoma (including low-grade, mantle cell, and intermediate-grade/diffuse) (Previously treated disease that has either relapsed or failed to respond adequately to conventional-dose therapy or autologous transplantation (AND) Chemoresponsive to most recent salvage therapy
* Multiple Myeloma (Presence of a poor risk cytogenetic abnormality \[i.e. 17p, t(4;14)\], Relapse post autologous transplant)

Exclusion Criteria:

* Poor cardiac function: left ventricular ejection fraction \<40%
* Poor pulmonary function: FEV1, FVC, or DLCO \<50% predicted
* Poor liver function: bilirubin \>2.5 mg/dl (not due to hemolysis, Gilbert's or primary malignancy), AST/ALT \> 3X ULN
* Poor renal function: Creatinine \>2.0 mg/dl or creatinine clearance (calculated creatinine clearance is permitted) \< 40 mL/min
* Ongoing or active systemic infection, active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive.
* Women of childbearing potential who currently are pregnant or who are not practicing adequate contraception
* Patients who have any debilitating medical or psychiatric illness which would preclude their giving informed consent or their receiving optimal treatment and follow-up.
* Systemic treatment, within 14 days before the first dose of MLN9708, with strong strong inhibitors of CYP3A (clarithromycin, telithromycin, itraconazole, voriconazole, ketoconazole, nefazodone, posaconazole) or strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of Ginkgo biloba or St. John's wort.
* Patient has \>/= Grade 3 peripheral neuropathy, or Grade 2 with pain on clinical examination during the screening period.
* Participation in other clinical trials, including those with other investigational agents not included in this trial, within 21days of the start of this trial and throughout the duration of this trial.
* Infection requiring systemic antibiotic therapy or other serious infection within 14 days before study enrollment.
* Evidence of current uncontrolled cardiovascular conditions, including uncontrolled hypertension, uncontrolled cardiac arrhythmias, symptomatic congestive heart failure, unstable angina, or myocardial infarction within the past 6 months.
* Known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent.
* Known GI disease or GI procedure that could interfere with the oral absorption or tolerance of MLN9708 including difficulty swallowing