Mobility has become a basic premise of network communications, thereby requiring a native integration into 5G networks. Despite numerous efforts to propose and standardize effective mobility-management models for IP, the result is a complex, poorly flexible set of mechanisms. The natural support for mobility offered by ICN (Information Centric Networking) makes it a good candidate to define a radically new solution relieving limitations of the traditional approaches. If consumer mobility is supported in ICN by design, in virtue of its connectionless pull-based communication model, producer mobility is still an open challenge. In this work, we look at two prominent ICN architectures, CCN (Content Centric Networking) and NDN (Named Data Networking) and we propose MAP-Me, an anchor-less solution to manage micro-mobility of content producers via a name-based CCN/NDN data plane, with support for latency-sensitive streaming applications. We analyze MAP-Me performance and provide guarantees of correctness, stability, and bounded stretch, which we verify on real ISP topologies. Finally, we set up a comprehensive simulation environment in NDNSim 2.1 for MAP-Me evaluation and comparison against the existing classes of solutions, including a realistic trace-driven car-mobility pattern under a 802.11n radio access. The results are encouraging and highlight the superiority of MAP-Me in terms of user performance and network cost metrics. All the code is available as open-source.