This paper addresses the problem of cross-register generalization in argument mining within political discourse. We examine whether models trained on adversarial, spontaneous U.S. presidential debates can generalize to the more diplomatic and prepared register of UN Security Council (UNSC) speeches. To this end, we conduct a comprehensive evaluation across four core argument mining tasks. Our experiments show that the tasks of detecting and classifying argumentative units transfer well across registers, while identifying and labeling argumentative relations remains notably challenging, likely due to register-specific differences in how argumentative relations are structured and expressed. As part of this work, we introduce ArgUNSC, a new corpus of 144 UNSC speeches manually annotated with claims, premises, and their argumentative links. It provides a resource for future in- and cross-domain studies and novel research directions at the intersection of argument mining and political science.