In an edge-cloud multi-tier network, datacenters provide services to mobile users, with each service having specific latency constraints and computational requirements. Deploying such a variety of services while matching their requirements with the available computing resources is challenging. In addition, time-critical services may have to be migrated as the users move, to keep fulfilling their latency constraints. Unlike previous work relying on an orchestrator with an always-updated global view of the available resources and the users' locations, this work envisions a distributed solution to the above problems. In particular, we propose a distributed asynchronous framework for service deployment in the edge-cloud that increases the system resilience by avoiding a single point of failure, as in the case of a central orchestrator. Our solution ensures cost-efficient feasible placement of services, while using negligible bandwidth. Our results, obtained through trace-driven, large-scale simulations, show that the proposed solution provides performance very close to those obtained by state-of-the-art centralized solutions, and at the cost of a small communication overhead.