Every year, millions of birds migrate between breeding and nonbreeding habitat, but the relative numbers of animals moving between sites are difficult to observe directly. Here we propose FlywayNet, a discrete network model based on observed count data, to determine the most likely migration links between regions using statistical modelling and efficient inference tools. Our approach advances on previous studies by accounting for noisy observations and flexible stopover durations by modelling using interacting hidden semi-Markov Models. In FlywayNet, individual birds sojourn in stopover nodes for a period of time before moving to other nodes with an unknown probability that we aim to estimate. Exact estimation using existing approaches is not possible, so we designed customised versions of the Monte Carlo expectation-maximisation and approximate Bayesian computation algorithms for our model. We compare the efficiency and quality of estimation of these approaches on synthetic data and an applied case study. Our algorithms performed well on benchmark problems, with low absolute error and strong correlation between estimated and known parameters. On a case study using citizen science count data of the Far Eastern Curlew (Numenius madagascariensis), an endangered shorebird from the East Asian-Australasian Flyway, the ABC and MCEM algorithms generated contrasting recommendations due to a difference in optimisation criteria and noise in the data. For ABC, we recovered key features of population-level movements predicted by experts despite the challenges of noisy unstructured data. Understanding connectivity places local conservation efforts and threat mitigation in the global context, yet it has proven difficult to rigorously quantify connectivity at the population level. Our approach provides a flexible framework to infer the structure of migratory networks in birds and other organisms.