Abemaciclib in Children With DIPG or Recurrent/Refractory Solid Tumors

Inclusion Criteria for All Participants: * Patient must have measurable or evaluable disease. * Age must be ≥ 2 years and \< 25 years * Body surface area (BSA) ≥ 0.5 m\^2 * Lansky (for participants ≤ 16 years) or Karnofsky (for participants \> 16 years) performance score ≥ 40 at the time of study enrollment * Adequate organ function at the time of study enrollment as follows: * Bone marrow: Absolute neutrophil count (ANC) ≥ 1,000/μL, platelet count ≥ 75,000/μL (transfusion independent for ≥ 7 days), hemoglobin concentration ≥ 8g/dL (may be transfused) * Patients with bone marrow metastatic disease who do not meet the above criteria will be eligible to enroll in the study with the following count criteria. These patients will not be evaluable for hematologic toxicity or hematologic DLT. * ANC \> 750/μL within 7 days prior to first dose of abemaciclib * Platelet count \> 50,000/μL (may receive platelet transfusions) within 7 days prior to first dose of abemaciclib * Hemoglobin ≥ 7.5 g/dL (may receive red blood cell (RBC) transfusions) within 7 days prior to first dose of abemaciclib * Renal: Normal serum creatinine concentration based on age or glomerular filtration rate (GFR) \> 70 ml/min/1.73m\^2 * Hepatic: Total bilirubin concentration \< 1.5x the institutional upper limit of normal for age; serum glutamic pyruvic transaminase (SGPT) \< 10x the institutional upper limit of normal for patients on Stratum A. Stratum B patients must have SGPT \< 4x the in…