As far as we know, no one is running Duloc; it's not clear if the city is operating autonomously, or got absorbed into a larger kingdom, or what.
The only hint we ever get about Duloc's fate comes in the TV special "Scared Shrekless", which is set after the events of the fourth movie, Shrek Forever After. The TV special is primarily set in Duloc castle, which is completely abandoned. There does not appear to be any Lord of Duloc, but we don't really see the town itself to know any more than that.
Given that Farquaad is only a Lord, he must have been a vassal to a larger kingdom. (Lords only get that way if a King gives them lands and titles). It's possible he swore allegiance to the King of Far Far Away, but we also know there's at least one other king (King Midas is mentioned in Shrek Forever After. Since Farquaad died without any children, that we know of, the most likely answer is that his lands reverted to that King and no new Lord was ever appointed.