Preheat the board.
Use a powerful soldering iron (temp controlled units are usually >60W).
Set the iron for a modest temperature. Not so high that it burns the board (and the solder, and the flux, and the tip), but probably above nominal (which should be around 350C for leaded solder).
Tin the joint as well as you can.
The last one seems to slip peoples' mind, especially when using a desoldering iron. Well, why would you want to add solder just to remove it again? Because you can't remove anything if you can't get it hot enough, and you need a clean, wet (maybe even saturated and globby) pool to conduct heat. And a properly tinned tip to swim in that pool. Whether you're using a regular pencil iron, or a hollow desoldering tip, or anything else, you must always have a shiny, sticky, wet, tinned surface.
In the end, it may turn out that it's simply better to rip the leads off and solder onto the nubs, rather than joust with a multilayer board. It can be very difficult to get a clean hole -- for example, it's pretty bad if you have to preheat the board so much that chip components start falling off, and electrolytics swell and burst, and so on...
Braid is probably less useful on a multilayer board, because the joint becomes less conductive as the solder gets sucked out. So it can be partially or fully frozen on the opposite side, the wick sucks up the melted region by the iron, and now you're left with a hollow, one-side-capped hole that doesn't carry enough heat through.
Tim