Climate change is increasing the frequency and severity of natural disasters worldwide. Media coverage of these events may be vital to generate empathy and mobilize global populations to address the common threat posed by climate change. Using a dataset of 466 news sources from 123 countries, covering 135 million news articles since 2016, we apply an event study framework to measure cross-border media activity following natural disasters. Our results shows that while media attention rises after disasters, it is heavily skewed towards certain events, notably earthquakes, accidents, and wildfires. In contrast, climatologically salient events such as floods, droughts, or extreme temperatures receive less coverage. This cross-border disaster reporting is strongly related to the number of deaths associated with the event, especially when the affected populations share strong social ties or genetic similarities with those in the reporting country. Achieving more balanced media coverage across different types of natural disasters may be essential to counteract skewed perceptions. Further, fostering closer social connections between countries may enhance empathy and mobilize the resources necessary to confront the global threat of climate change.