Anecdotally, I have not observed an increase in empathy from people who immerse themselves in negative news about faraway places. There's sometimes an increase in the performance of empathy—you'll see them bemoan the various crises, you'll see them wonder aloud how anyone can go about their lives while ${STUFF} is happening—but I haven't observed an increase in their ability to actually be empathetic to the perspectives and needs of people closer to home.
If anything it's the opposite: the more immersed someone is in tragedies and crises on the other side of the planet, the more aggressive and thoughtless they are in response to hot topics locally. Instead of slowing down and developing empathy for those who disagree with them, they lash out. It's like being immersed in extreme crises they can't control causes them to perceive their local crises as more severe than they actually are and triggers fight or flight, which actually stands in the way of empathy.