Enriched finite element methods such as the Generalized Finite Element Method (GFEM), the eXtended Finite Element Method (XFEM), the cut-cell finite element method, the Interface-enriched Generalized Finite Element Method (IGFEM), and the Conforming Decomposition Finite Element Methods (CDFEM)[1] are powerful tools for coupled multiphase and multimaterial problems with moving interfaces. To capture the discontinuities across interfaces, these methods introduce some form of enrichment of the degrees of freedom. Additional unknowns are assigned to the mesh entities (elements, nodes, sides, or edges) that are associated with these interfacial elements, and additional equations are formulated. In CDFEM, level sets are used to describe the domain of each material or phase. Nodes are added at the intersection of each level set surface with the edges of the input mesh, and a conforming mesh is generated automatically. This allows the weak and strong discontinuities across the interfaces to be captured using standard finite element methods. In recent work, a new strategy has been developed for automated interface conforming tetrahedral mesh generation, which produces higher quality meshes than standard CDFEM techniques. The method is termed the Conforming Transient h-r Unstructured Adaptive Mesh Refinement (cThruAMR). cThruAMR uses a combination of h-adaptivity and r-adaptivity to generate high quality meshes that conform to a moving interface. The term h-adaptivity is used for refining or cutting the mesh. Conforming h-adaptivity is used in CDFEM to capture dynamic topology problems. The term r-adaptivity is used for moving the nodes of the mesh to capture a desired feature. By combining h and r adaptivity, cThruAMR can produce high quality meshes even for dynamic topology problems. The method is closely related to the Conforming to Interface Structured Adaptive Mesh Refinement (CISAMR) method but employs a general unstructured mesh and is developed for transient moving interface problems. This focus of this talk is the description of cThruAMR for capillary hydrodynamics including the strategies employed for r-adaptivity, level set advection, dynamic DOF handling, and interfacial boundary conditions. The method is verified using multiple benchmark problems and the method is shown to produce optimal rates of convergence. The resulting discretization is shown to have better conditioning and improved mesh quality compared to standard CDFEM discretizations.