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Automated classification of building damage in remote sensing images enables the rapid and spatially extensive assessment of the impact of natural hazards, thus speeding up emergency response efforts. Convolutional neural networks (CNNs) can reach good performance on such a task in experimental settings. How CNNs perform when applied under operational emergency conditions, with unseen data and time constraints, is not well studied. This study focuses on the applicability of a CNN-based model in such scenarios. We performed experiments on 13 disasters that differ in natural hazard type, geographical location, and image parameters. The types of natural hazards were hurricanes, tornadoes, floods, tsunamis, and volcanic eruptions, which struck across North America, Central America, and Asia. We used 175,289 buildings from the xBD dataset, which contains human-annotated multiclass damage labels on high-resolution satellite imagery with red, green, and blue (RGB) bands. First, our experiments showed that the performance in terms of area under the curve does not correlate with the type of natural hazard, geographical region, and satellite parameters such as the off-nadir angle. Second, while performance differed highly between occurrences of disasters, our model still reached a high level of performance without using any labeled data of the test disaster during training. This provides the first evidence that such a model can be effectively applied under operational conditions, where labeled damage data of the disaster cannot be available timely and thus model (re-)training is not an option.