The potential impact of automation on the labor market is a topic that has generated significant interest and concern amongst scholars, policymakers, and the broader public. A number of studies have estimated occupation-specific risk profiles by examining the automatability of associated skills and tasks. However, relatively little work has sought to take a more holistic view on the process of labor reallocation and how employment prospects are impacted as displaced workers transition into new jobs. In this paper, we develop a new data-driven model to analyze how workers move through an empirically derived occupational mobility network in response to automation scenarios which increase labor demand for some occupations and decrease it for others. At the macro level, our model reproduces a key stylized fact in the labor market known as the Beveridge curve and provides new insights for explaining the curve's counter-clockwise cyclicality. At the micro level, our model provides occupation-specific estimates of changes in short and long-term unemployment corresponding to a given automation shock. We find that the network structure plays an important role in determining unemployment levels, with occupations in particular areas of the network having very few job transition opportunities. Such insights could be fruitfully applied to help design more efficient and effective policies aimed at helping workers adapt to the changing nature of the labor market.