Future networks are expected to support lowlatency, context-aware and user-specific services in a highly flexible and efficient manner. One approach to support emerging use cases such as, e.g., virtual reality and in-network image processing is to introduce virtualized network functions (vNF)s at the edge of the network, placed in close proximity to the end users to reduce end-to-end latency, time-to-response, and unnecessary utilisation of the core network. While placement of vNFs has been studied before, it has so far mostly focused on reducing the utilisation of server resources (i.e., minimising the number of servers required in the network to run a specific set of vNFs), and not taking network conditions into consideration such as, e.g., end-to-end latency, the constantly changing network dynamics, and user mobility patterns. In this paper, we first formulate the Edge vNF placement problem to allocate vNFs to a distributed edge infrastructure, minimising end-to-end latency from all users to their associated vNFs. Furthermore, we present a way to dynamically reschedule the optimal placement of vNFs based on temporal network-wide latency fluctuations using optimal stopping theory. We evaluate our dynamic scheduler over a simulated nationwide backbone network using real-world ISP latency characteristics. We show that our proposed dynamic placement scheduler minimises vNF migrations compared to other schedulers (e.g., periodic and always-on scheduling of a new placement), and offers Quality of Service guarantees by not exceeding a maximum number of latency violations that can be tolerated by certain applications.