When they stalk/follow/chase after them in hopes of catching photos or stories to sell. Pursue: follow (someone or something) in order to catch or attack them. "the officer pursued the van"

Pursue means - follow (someone or something) in order to catch or attack them. The police pursue the robber. The lions pursue the gazelle. Reporters pursue famous people by following them around and taking pictures of them constantly. A member of the British Royal Family (Lady Diana) died because of this constant pursuit by the media.