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We consider a hierarchical edge-cloud architecture in which services are provided to mobile users as chains of virtual network functions. Each service has specific computation requirements and target delay performance, which require placing the corresponding chain properly and allocating a suitable amount of computing resources. Furthermore, chain migration may be necessary to meet the services' target delay, or convenient to keep the service provisioning cost low. We tackle such issues by formalizing the problem of optimal chain placement and resource allocation in the edge-cloud continuum, taking into account migration, bandwidth, and computation costs. Specifically, we first envision an algorithm that, leveraging resource augmentation, addresses the above problem and provides an upper bound to the amount of resources required to find a feasible solution. We use this algorithm as a building block to devise an efficient approach targeting the minimum-cost solution, while minimizing the required resource augmentation. Our results, obtained through trace-driven, large-scale simulations, show that our solution can provide a feasible solution by using half the amount of resources required by state-of-the-art alternatives.