In addition to what Dark Panther already mentioned, "ran down" can also indicate a direction of running (if it's downhill or perhaps if the destination is south of where you started (though not necessarily south). Also used if you're coming from a tall building and you had to go to somewhere on the street level, that would also be used.
"I ran down to the post office before it closed."
"The boy ran down to the piers to watch the ship come in."
"Run down" can also mean to be hit by a car (or other vehicle)! :)
"The woman got run down by the bus."
Best,
~_~. Maidhc