MOSAIC constructs new overlay networks with desired characteristics by composing existing overlays with subsets of those attributes. Thus, MOSAIC overcomes the problem of multiple network infrastructures that are partial solutions, while preserving deployability. Composition of control and/or data planes is possible in the system. MOSAIC overlays are specified in Mozlog, a declarative language that specifies overlay properties without binding them to a particular implementation or underlying network. This paper focuses on the runtime aspects of MOSAIC: how it enables interoperability between different overlay networks and how it implements switching between different overlay compositions, permitting dynamic compositions with both existing overlay networks and legacy applications. The system is validated experimentally using declarative overlay compositions concisely specified in Mozlog: an indirection overlay that supports mobility (i3), a resilient overlay (RON), and scalable lookups (Chord), all of which are combined to provide new functionality. MOSAIC provides the benefits of runtime composition to simultaneously deliver application-aware mobility, NAT traversal and reliability with low performance overhead, demonstrated by measurements on both a local cluster and PlanetLab.