A challenge for practitioners of Bayesian inference is specifying a single model that incorporates multiple heterogeneous data. It may be easier instead to specify distinct submodels for each source of data, then "join" the submodels together. When all submodels contain the same common quantity, which could be a parameter or deterministic function thereof, the submodels can be joined using Markov melding. However, it is unclear how to join submodels when they are linked in a chain structure, i.e. where neighbouring submodels have quantities in common. We propose chained Markov melding as a method for combining chains of submodels into a joint model, and demonstrate our methodology using two examples. The first example is an integrated population model used in conservation ecology, where multiple data are required to accurately estimate population immigration and reproduction rates. We also consider a joint longitudinal and time-to-event model with the added complication that event times are derived from a separate submodel, and hence are uncertain. We show that failing to incorporate uncertainty in the event times results in overconfident and biased posterior inference.