Understanding how the composition of guest origin markets evolves over time is critical for destination marketing organizations, hospitality businesses, and tourism planners. We develop and apply Bayesian Dirichlet autoregressive moving average (BDARMA) models to forecast the compositional dynamics of guest origin market shares using proprietary Airbnb booking data spanning 2017--2025 across four major destination regions. Our analysis reveals substantial pandemic-induced structural breaks in origin composition, with heterogeneous recovery patterns across markets. In our analysis, the BDARMA framework achieves the lowest forecast error for EMEA and competitive performance across destination regions, outperforming standard benchmarks including naïve forecasts, exponential smoothing, and SARIMA on log-ratio transformed data in compositionally complex markets. For EMEA destinations, BDARMA achieves 27% lower forecast error than naïve methods ($p < 0.001$), with the greatest gains where multiple origin markets compete in the 5-25% share range. By modeling compositions directly on the simplex with a Dirichlet likelihood and incorporating seasonal variation in both mean and precision parameters, our approach produces coherent forecasts that respect the unit-sum constraint while capturing complex temporal dependencies. The methodology provides destination stakeholders with probabilistic forecasts of source market shares, enabling more informed strategic planning for marketing resource allocation, infrastructure investment, and crisis response.

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