Live migration of virtual machines (VMs) is key feature of virtualization that is extensively leveraged in IaaS cloud environments: it is the basic building block of several important features, such as load balancing, pro-active fault tolerance, power management, online maintenance, etc. While most live migration efforts concentrate on how to transfer the memory from source to destination during the migration process, comparatively little attention has been devoted to the transfer of storage. This problem is gaining increasing importance: due to performance reasons, virtual machines that run large-scale, data-intensive applications tend to rely on local storage, which poses a difficult challenge on live migration: it needs to handle storage transfer in addition to memory transfer. This paper proposes a memory-migration independent approach that addresses this challenge. It relies on a hybrid active push / prioritized prefetch strategy, which makes it highly resilient to rapid changes of disk state exhibited by I/O intensive workloads. At the same time, it is minimally intrusive in order to ensure a maximum of portability with a wide range of hypervisors. Large scale experiments that involve multiple simultaneous migrations of both synthetic benchmarks and a real scientific application show improvements of up to 10x faster migration time, 10x less bandwidth consumption and 8x less performance degradation over state-of-art.