Quickest way I've found:
1) Get roughly in the zone for a new move out request through normal TT'ing (I do 3 days forward, 1 day back first), save, quit, go to system menu.
2) Change the date on 3DS system through the System Settings menu to 9 days ahead, reboot ACNL, start new character, check the map. If someone is gone, that's your mover.
3) If you've found the mover through this method and don't want them to leave, drop back to 3DS system menu without saving, change date to the original (where you started), restart with your mayor and target that villager. From my experience, they will now immediately tell you that they are thinking about moving.
If no one is gone, it means that you have not triggered a villager move out yet. In that case, drop back to system menu without saving, change 3DS system time to the original date, continue your game with the mayor, TT one day forward with ACNL time settings. Repeat the above process.
If the villager gone is who you want to be gone, you can just finish the new move in process (isabelle, nook, isabelle) and save to finalize it. Or you can return to menu, start with mayor, target that character, etc. at your preference.
Honestly, I spent hours and hours trying to do the ping method, and while the above is boring, from my experience it is much faster overall as you are not waiting for random pings and getting all of the other villager requests for fruit and nicknames, etc in the meantime. It also makes it so that you don't need to keep save & quit to see all of the villagers outside or talk to everyone 9,000 times to try and find rumors.
It can still take forever and a day to get the right villager to move out, but is the lesser of two evils imo.
A couple additional comments:
1) Plant flowers on your mud spots.
You're TT'ing like crazy, take advantage of it. Grass will regrow pretty fast if you have flowers on the spots and water them occasionally on stop-offs.
2) Leave a shovel for your new character
If you don't have many spaces that the new character can save in (I only have one plot in my village big enough for a new house), leave a shovel somewhere that he can get it, since sometimes the random gem rock will form on the spot you need to build your new house on. For some reason, Nook won't let the new character build a home on a gem rock, even though it's fake. Use the shovel to break it and he won't care.
