To chase and capture someone or something. The police managed to run down the suspect after a long chase.
To criticize someone or something in a disparaging manner. She tends to run down her colleagues when they are not around.
To gradually lose power or stop functioning. The battery will run down if you leave the flashlight on all night.
To hit and knock down a person or animal with a vehicle. The driver was arrested after he ran down a pedestrian at the crosswalk.
To trace or track down something or someone. I need to run down the source of this information before we publish it.