Testing the Addition of an Anti-cancer Drug, Abemaciclib, to the Usual Chemotherapy Treatment (Gemcitabine) for Soft Tissue Sarcoma

Inclusion Criteria: * Phase 1: Patients must have advanced/metastatic histologically confirmed soft tissue sarcoma and have received at least one prior standard systemic therapy (prior gemcitabine is allowed) * Phase 2: Patients must have advanced/metastatic pathologically confirmed leiomyosarcoma or dedifferentiated liposarcoma for which gemcitabine and docetaxel is considered standard-of-care, patients may be systemic-treatment naïve. Prior gemcitabine is not allowed * Patients must have presence of measurable/assessable tumor * Patients must have intact Rb gene expression in the baseline tumor biopsy or archived tumor sample, as assessed by immunohistochemistry (at MD Anderson: clone G3- 245, BD Pharmagen, RRID:AB\_385259, Clinical Laboratory Improvement Act \[CLIA\] certified antibody) * Age ≥ 18 years. Because no dosing or adverse event data are currently available on the use of abemaciclib in combination with gemcitabine in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative oncology Group (ECOG) performance status ≤ 2 (Karnofsky ≥ 60%) * Absolute neutrophil count ˃ 1.2K/µL * Hemoglobin ˃ 9.0 g/dL * Platelets ˃ 100K/mm\^3 * Glomerular filtration rate (GFR) ≥ 60 mL/min unless data exists supporting safe use at lower kidney function values, no lower than 30 mL/min * Total bilirubin ≤ 1.5 × institutional upper limit of normal (ULN), patient with Gilbert's syndrome ≤ 2.0 times ULN, or direct bilirubin within normal limits * Aspartate am…