Mate the connector with a spare one of the opposite gender to support the pins while they are hot. Use the biggest bit that you can get onto a single pin at a time, and a high wattage iron, as you need rapid heat transfer while the wire's still in there. Capt bullshot has described the general technique. Desolder braid is too slow and suckers perform poorly on a blind hole so they aren't much use for connector cleanup. You can try heating the terminal and rapping the whole connector body on the edge of a shallow tin so the solder spatter goes in the tin rather than everywhere across your bench. However if the original joint was good, there will be about the right amount of solder left to make a new joint with a freshly tinned and fluxed wire end, by simply pushing it in as soon as the terminal reflows, and only adding extra solder if the fillet is deficient.