Motivated by the challenge of analyzing the dynamics of weekly sea border crossings in the Mediterranean (2015–2025) and the English Channel (2018–2025), we develop a Bayesian dynamic framework for modeling heteroskedastic count time series. Building on theoretical considerations and empirical stylized facts, our approach utilizes a Poisson random walk model that allows for heavy-tailed innovations or stochastic volatility dynamics, while incorporating an explicit mechanism to separate structural from sampling zeros. Posterior inference is carried out via a straightforward Markov chain Monte Carlo algorithm. Applying this methodology to the Mediterranean and English Channel data, we compare alternative model specifications through a comprehensive out-of-sample density forecasting exercise. Evaluating each model using log predictive scores and empirical coverage up to the 99th percentile, we find strong evidence for stochastic volatility in the migration innovations, with these models producing well-calibrated forecasts even at extreme quantiles. Our framework can be used to develop risk indicators with direct policy implications for improving governance and preparedness for migration surges. More broadly, the methodology extends to other zero-inflated non-stationary count time series applications, including epidemiological surveillance and public safety incident monitoring.