Functional linear models have been a staple in the analysis of regression relationships between a functional response and a functional predictor. Since many functional data sets require taking into account serial dependence, it is important to include a time series component in the model building process. This paper discusses such an approach in the context of structural break analysis, which may be useful whenever it is doubtful if a regression relationship remains stable over time. The model studied here is a dependent functional regression, with time series structures enabled through imposing weak dependence assumptions. Estimation in this model is performed based on dimension reduction of both response and predictor function. The resulting multivariate linear model is estimated with ordinary least squares. The main theoretical result establishes the large-sample behavior of this estimator, which is shown to be biased even in the limit. The theory helps guide the construction of a test for the presence of structural breaks. The finite sample properties of this test are evaluated through simulations and an application to environmental data.