The Multi-access Edge Computing (MEC) and Fog Computing paradigms are enabling the opportunity to have middleboxes either statically or dynamically deployed at network edges acting as local proxies with virtualized resources for supporting and enhancing service provisioning in edge localities. However, migration of edge-enabled services poses significant challenges in the edge computing environment. In this paper, we propose an edge computing platform architecture that supports service migration with different options of granularity (either entire service/data migration, or proactive application-aware data migration) across heterogeneous edge devices (either MEC-based servers or resource-poor Fog devices) that host virtualized resources (Docker Containers). The most innovative elements of the technical contribution of our work include i) the possibility to select either an application-agnostic or an application-aware approach, ii) the possibility to choose the appropriate application-aware approach (e.g., based on data access frequencies), iii) an automatic edge services placement support with the aim of finding a more effective placement with low energy consumption, and iv) the in-lab experimentation of the performance achieved over rapidly deployable environments with resource-limited edges such as Raspberry Pi devices.