Participants who have previously received a cyclin-dependent kinase (CDK) 4/6 inhibitor For Part I (abemaciclib + endocrine therapy): The participant must have demonstrated evidence of disease progression on a Cyclin Dependent Kinase 4 (CDK4) and Cyclin Dependent Kinase 6 (CDK6) inhibitor (either palbociclib or ribociclib) plus endocrine therapy for advanced or metastatic disease as the most recent therapy immediately preceding study entry. The participant should remain on the current endocrine therapy while receiving abemaciclib. Prior therapy with any cyclin-dependent kinase (Cdk) 4 inhibitor one prior hormonal therapy and a Cyclin-dependent kinase (CDK)4/6 inhibitor. Note: Participants may have received treatment for brain metastases, but must be neurologically stable, completed radiotherapy and off corticosteroids for at least one month prior to starting trial therapy. Prior therapy with a cyclin-dependent kinase (CDK) inhibitor History of exposure to alvocidib or any other cyclin-dependent kinase 9 (CDK9) inhibitor. Arm B: MBC with progression during or following one prior endocrine based systemic therapy in the metastatic setting, with no prior therapy with any cyclin-dependent kinase (CDK) inhibitor; Prior therapy with any cyclin-dependent kinase (CDK) inhibitor Prior treatment with any cyclin dependent kinase (CDK) inhibitor Has been previously treated with a cyclin-dependent kinase (CDK) inhibitor For Parts A, B, D, E, and F: Have previously received treatment with any cyclin dependent kinase 6 (CDK6) inhibitor. For Part C participants may have received prior palbociclib or ribociclib, but not abemaciclib treatment. Prior treatment with drug targeting cyclin-dependent kinase (CDK) family Prior cyclin-dependent kinase (CDK)4/6 inhibitor exposure received prior treatment with any cyclin-dependent kinase (CDK) 4 and CDK 6 inhibitor Have previously received treatment with any cyclin-dependent kinase (CDK) 4 and 6 inhibitor or phosphatidylinositol 3-kinase (PI3K) and/or mammalian target of rapamycin (mTOR) inhibitor or have a known hypersensitivity to any component of the investigational products in this study. Have received prior treatment with any cyclin dependent kinase (CDK) 4 and 6 inhibitor or participated in a clinical trial with a CDK 4 and 6 inhibitor and the treatment administered is not known. Have received prior treatment with any cyclin-dependent kinase (CDK) 4/6 inhibitor (or participated in any CDK4/6 inhibitor clinical trial for which treatment assignment is still blinded) Disease that progressed during treatment or within 1 month after the end of treatment with prior tamoxifen, AI, or cyclin-dependent kinase (CDK) 4/6 inhibitor plus letrozole, for advanced/metastatic disease. Have received treatment with a prior cyclin-dependent kinase (CDK4) and CDK 6 inhibitor. Prior therapy with any cyclin-dependent kinase 4/6 (CDK4/6) inhibitor Prior systemic anticancer therapy for SCLC (including previous therapy with a cyclin-dependent kinase [CDK] inhibitor) Patients who have received prior treatment with a selective cyclin-dependent kinase 4/6 (CDK4/6) inhibitor Stage I Arm B: history of significant toxicity related to cyclin-dependent kinase (CDK)4/6 inhibitor, bone marrow transplant or extensive radiotherapy to ?25 percent (%) of bone marrow Stage II: