Automated stance detection and related machine learning methods can provide useful insights for media monitoring and academic research. Many of these approaches require annotated training datasets, which limits their applicability for languages where these may not be readily available. This paper explores the applicability of large language models for automated stance detection in a challenging scenario, involving a morphologically complex, lower-resource language, and a socio-culturally complex topic, immigration. If the approach works in this case, it can be expected to perform as well or better in less demanding scenarios. We annotate a large set of pro and anti-immigration examples, and compare the performance of multiple language models as supervised learners. We also probe the usability of ChatGPT as an instructable zero-shot classifier for the same task. Supervised achieves acceptable performance, and ChatGPT yields similar accuracy. This is promising as a potentially simpler and cheaper alternative for text classification tasks, including in lower-resource languages. We further use the best-performing model to investigate diachronic trends over seven years in two corpora of Estonian mainstream and right-wing populist news sources, demonstrating the applicability of the approach for news analytics and media monitoring settings, and discuss correspondences between stance changes and real-world events.