Lenvatinib and Capecitabine in Patients With Advanced Malignancies

Study Groups: If participant is are found to be eligible to take part in this study, they will be assigned to a study group based on when they join this study. Up to 4 groups of up to 18 participants will be enrolled in Phase 1 of the study, and up to 28 participants will be enrolled in Phase 2. If participant is enrolled in Phase 1, the dose of lenvatinib they receive will depend on when they join this study. The first group of participants will receive the lowest dose level of lenvatinib. Each new group will receive a higher dose of lenvatinib than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose of lenvatinib is found. If participant is enrolled in Phase 2, they will receive lenvatinib at the highest dose that was tolerated in Phase 1. All participants will receive the same dose of capecitabine. Study Drug Administration: Each study cycle is 21 days. Participant will take lenvatinib by mouth every day and capecitabine by mouth 2 times every day. Length of Study Participation: Participant may continue taking the study drugs for as long as the doctor thinks it is in their best interest. Participant will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if they are unable to follow study directions. Patient's participation on the study will be over after the end-of-study visit (described below). Study Visits: During Week 2 of Cycle 1: * Participant will have a physical exam. * Blood (about 3 teaspoons) will be drawn for routine tests. * If participant is enrolled in Phase 2, they will have a tumor biopsy for biomarker testing. * If participant is enrolled in Phase 2, blood (about 2 teaspoons) will be drawn for biomarker, CTC, and cfDNA testing. During Week 3 of Cycle 1, blood (about 3 teaspoons) will be drawn for routine tests. During Week 1 of Cycles 2 and beyond: * Participant will have a physical exam. * Blood (about 3 teaspoons) will be drawn for routine tests. * If participant is enrolled in Phase 2, blood (about 2 teaspoons) will be drawn for biomarker, CTC, and cfDNA testing. * If participant can become pregnant, blood (about 1 teaspoon) will be collected for a pregnancy test. During Weeks 2 and 3 of Cycles 2 and beyond, participant will be called by a member of the study staff and asked about any side effects they may have. This call should last about 5-10 minutes. During Week 3 of Cycle 2 and then every even-numbered cycle after that (Cycles 4, 6, 8, and so on), participant will have a CT scan or MRI. End-of-Study Visit: After participant's last dose of study drug, they will have an end-of-study visit. At this visit, the following tests and procedures will be performed: * Participant will have a physical exam. * Blood (about 3 teaspoons) will be drawn for routine tests. * Participant will have a CT scan or MRI to check the status of the disease. * If participant can become pregnant, blood (about 1 teaspoon) or urine will be collected for a pregnancy test. * If participant is enrolled in Phase 2, blood (about 2 teaspoons) will be drawn for biomarker, CTC, and cfDNA testing. This is an investigational study. Lenvatinib is FDA approved and commercially available for the treatment of certain types of thyroid cancer. Capecitabine is FDA approved and commercially available for the treatment of certain types of breast and colorectal cancers. It is considered investigational to use lenvatinib and capecitabine to treat advanced cancer. The study doctor can explain how the study drugs are designed to work. Up to 46 participants will be enrolled in this study. All will take part at MD Anderson.