Today, the volume of data in the world has been tremendously increased. Large-scaled and diverse data sets are raising new big challenges of storage, process, and query. Tiered storage architectures combining solid-state drives (SSDs) with hard disk drives (HDDs), become attractive in enterprise data centers for achieving high performance and large capacity simultaneously. However, how to best use these storage resources and efficiently manage massive data for providing high quality of service (QoS) is still a core and difficult problem. In this paper, we present a new approach for automated data movement in multi-tiered storage systems, which lively migrates the data across different tiers, aiming to support multiple service level agreements (SLAs) for applications with dynamic workloads at the minimal cost. Trace-driven simulations show that compared to the no migration policy, LMsT significantly improves average I/O response times, I/O violation ratios and I/O violation times, with only slight degradation (e.g., up to 6% increase in SLA violation ratio) on the performance of high priority applications.