MV-NIS Infected Mesenchymal Stem Cells in Treating Recurrent Ovarian, Primary Peritoneal or Fallopian Tube Cancer

Inclusion Criteria: * Must have: * Recurrent or progressive ovarian cancer, primary peritoneal cancer or fallopian tube cancer after prior treatment with platinum and taxanes * Histologic confirmation of the original primary tumor * Prior bilateral oophorectomy * The following histologic epithelial cell types are eligible: serous adenocarcinoma, endometrioid adenocarcinoma, mucinous adenocarcinoma, undifferentiated carcinoma, clear cell adenocarcinoma, mixed epithelial carcinoma, transitional cell carcinoma, malignant Brenner's tumor, or adenocarcinoma not otherwise specified (NOS) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, 2 * Absolute neutrophil count (ANC) \>= 1500/uL (obtained =\< 7 days prior to registration) * Platelet (PLT) \>= 100,000/uL (obtained =\< 7 days prior to registration) * Total bilirubin =\< upper normal limit (obtained =\< 7 days prior to registration) * Aspartate aminotransferase (AST) =\< 2 x upper limit of normal (ULN) (obtained =\< 7 days prior to registration) * Creatinine =\< 1.5 x ULN (obtained =\< 7 days prior to registration) * Hemoglobin (Hgb) \>= 9.0 g/dL (obtained =\< 7 days prior to registration) * Normal cardiac function as defined by a normal ejection fraction by MUGA (multi gated acquisition scan) or echocardiogram * Provide informed written consent * Willing to return to Mayo Clinic Rochester for follow-up * Life expectancy \>= 12 weeks * Willing to provide all biologic specimens as required by the protocol * Measurable disease by exam or CT scan, or for patients with cancer antigen (CA)-125 elevation or with microscopic residual but without measurable disease on imaging, willingness to undergo laparoscopy for evaluation of treatment effect if no radiographic progression after 6 treatment cycles * CD4 count \>= 200/uL or \>= 15% of peripheral blood lymphocytes Exclusion Criteria: * Epithelial tumors of low malignant potential, stromal tumors, and germ cell tumors of the ovary * Known standard therapy for the patient's disease that is potentially curative or definitely capable of extending life expectancy; subjects will be excluded if this is their first relapse and they have recurred \> 6 months from completion of primary (adjuvant) chemotherapy * Active infection =\< 5 days prior to registration * History of tuberculosis or history of tuberculosis skin test purified protein derivative (PPD) positivity * History of other malignancy =\< 5 years prior to registration except for non-melanoma skin cancer, carcinoma in situ of the cervix, and ductal carcinoma in situ (DCIS) * Any of the following prior therapies: * Chemotherapy =\< 3 weeks prior to registration * Immunotherapy =\< 4 weeks prior to registration * Biologic therapy =\< 4 weeks prior to registration * Extensive abdominal surgery if it includes enterotomy(ies) =\< 3 weeks prior to registration; this criterion does not apply to placement of the peritoneal Port-A-Cath or lysis of adhesions at the time of registration * Any viral or gene therapy prior to registration * Radiation therapy to the abdomen or pelvis * New York Heart Association classification III or IV, known symptomatic coronary artery disease, or symptoms of coronary artery disease on systems review, or known cardiac arrhythmias (atrial fibrillation or supraventricular tachycardia \[SVT\]) * Other cardiac or pulmonary disease that, at the investigators discretion, can impair treatment safety * Requiring blood product support * Central nervous system (CNS) metastases or seizure disorder * Human immunodeficiency virus (HIV)-positive test result or history of other immunodeficiency * History of organ transplantation * History of chronic hepatitis B or C * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-Food and Drug Administration \[FDA\]-approved indication and in the context of a research investigation) * Intra-abdominal disease \> 8 cm in diameter at the time of registration, intrahepatic disease, or disease beyond the abdominal cavity; patients with intra-abdominal lymph node involvement are eligible based on biodistribution data indicating viral dissemination to lymph nodes following intraperitoneal administration * Treatment with oral/systemic corticosteroids, with the exception of topical or inhaled steroids * Exposure to household contacts =\< 15 months old or household contact with known immunodeficiency * Allergy to measles vaccine or history of severe reaction to prior measles vaccination * Allergy to iodine; this does not include reactions to intravenous contrast materials * Any other pathology or condition where the principle investigator may deem to negatively impact treatment safety