Recent research has focused on continuous monitoring of nearest neighbors (NN) in highly dynamic scenarios, where the queries and the data objects move frequently and arbitrarily. All existing methods, however, assume the Euclidean distance metric. In this paper we study k-NN monitoring in road networks, where the distance between a query and a data object is determined by the length of the shortest path connecting them. We propose two methods that can handle arbitrary object and query moving patterns, as well as fluctuations of edge weights. The first one maintains the query results by processing only updates that may invalidate the current NN sets. The second method follows the shared execution paradigm to reduce the processing time. In particular, it groups together the queries that fall in the path between two consecutive intersections in the network, and produces their results by monitoring the NN sets of these intersections. We experimentally verify the applicability of the proposed techniques to continuous monitoring of large data and query sets.