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 aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\]) ≤ 1.5 × institutional ULN * Alanine aminotransferase (ALT)(serum glutamic pyruvic transaminase \[SGPT\]) ≤ 1.5 × institutional ULN * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association (NYHA) Functional Classification. To be eligible for this trial, patients should be class congestive heart failure (CHF) II or better * Patients must have a life expectancy of greater than 6 months * Females of childbearing potential must have a negative serum pregnancy test within one week of trial enrollment and be willing to use an adequate method of contraception to avoid pregnancy throughout the trial and for up to 6 months after the last dose of drug therapy. The effects of abemaciclib on the developing human fetus are unknown. For this reason and because CDK 4/6 inhibiting agents as well as gemcitabine are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of abemaciclib administration. Abstinence is considered an effective method only if defined as refraining from heterosexual intercourse during the entire period of risk associated with the study ranging from three weeks prior to initiation of treatment and up to 6 months after the last dose of treatment * Ability to understand and the willingness to sign a written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of study participants * Patient is capable of swallowing oral medications Exclusion Criteria: * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia * Patients who are receiving any other investigational agents. There must be no investigational drug use within 30 days or 5 half-lives of receiving the first dose of treatment on this treatment * History of allergic reactions attributed to compounds of similar chemical or biologic composition to abemaciclib or gemcitabine * Patients with uncontrolled intercurrent illness or any other significant condition(s) that would make participation in this protocol unreasonably hazardous * Pregnant women are excluded from this study because abemaciclib is a CDK4/6 inhibiting agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with abemaciclib or gemcitabine, breastfeeding should be discontinued if the mother is treated with abemaciclib or gemcitabine * Use of strong CYP34A inhibitors which cannot be discontinued by the patient prior to trial initiation. The washout period of these drugs should be 5 half-lives * Progression on prior CDK4 inhibitor therapy * Phase 2 only: Prior gemcitabine-based chemotherapy * Presence of significant cardiac disease. Significant cardiac disease includes personal history of any of the following conditions: syncope of cardiovascular etiology, ventricular arrhythmia of pathological origin (including, but not limited to, ventricular tachycardia and ventricular fibrillation), or sudden cardiac arrest * Patients with interstitial lung disease (ILD) * Patients with gastrointestinal conditions that may affect the absorption of oral medications * Patients must not have received or be scheduled to receive radiation therapy within 7 days or less from gemcitabine administration * Patients must not have had major surgery within 14 days prior to randomization