Justisaur:
I don't think it's all that much of a personal thing. It seems to be something every game both run by me and I've been in struggles with. I think that having guidance from GMs of games that have been running a long time and deal with this issue well is invaluable, it's perhaps the single most important thing one can know about running a game PbP.
I don't find answers of 'do it your own way' helpful, when your own way isn't working.
Sorry you didn't find my reply, and that of others, to be helpful. I'll try again, cos I want to help. :)
Firstly, this is something that
every game struggles with, and there is no formula answer. If there were, we wouldn't still struggle with it. That's what we mean by 'do it your way'.
I'm running one game that is over 6000 posts, one that is over 9000 posts, and co-gming one that is over 10,000 posts, but I can't claim to be 'getting it right', because my games still suffer with slow posts, long absences and drop outs. My way isn't working, either. Nobody has got this licked.
Quite often, there are only a couple of the founding players left in games like these. I doubt if any game reaches five figures with its original cast intact. Players come and go, but the show goes on.
All you can do is steer a course that you think is most likely to keep the game going, regardless of the individual characters.
To answer your specific questions:
Do you have any rules about moving things along when people don't post?
Yes. But then I break them all within a few weeks of the game starting, because no plan survives contact with the players. Generally I just play it by ear. What works for one game doesn't work for an other, even with the same GM and some of the same players. If some players drop out and other, different people join, the whole game dynamic might change, and you have to adapt to the new situation.
Do the PCs get turned into NPCs temporarily, or disappear, when their owners don't post?
Depends on what's good for the game. You may have a game in which characters can simply walk away, or get killed without their absence destroying the party, but if not, you'll need to NPC the characters until it's convenient to dump them.
How long do you or your GM let people go without posting before replacing them?
Another piece of string. Again, whatever is best for the game. If you have six players and five of them stop posting for five different reasons, you might decide to put the game on hold for a while and see who comes back.
If five of your players are posting regularly and the sixth keeps disappearing without notice for days or weeks on end, you might decide to replace him sooner rather than later.
Things happen to people. Yes, you
could log in and post a message to your five different GMs to tell them all that your kid's sick. But maybe you've got other things on your mind.
Usually, I'll post an absence query after a week, or two, depending what's going on in the game and in my life, and how distracted I am, and I'll hope to get a reply within a day or two. Depending on what I hear back, I may wait a while longer, or I may drop the character, or NPC it or whatever I feel is appropriate.
I'm especially interested in hearing from anyone in or running long running games, as I haven't been in any games that have lasted terribly long, my own have been longer than others I've played at, but eventually it gets frustrating trying to deal with this and I give up on the games.
My games are reasonably long running, but the only way to make that happen is to keep banging your head against that brick wall until the wall cracks. Yes it's frustrating, and it will continue to be frustrating right through to the end.
The long games last not because players behave themselves and post promptly in response to a magic formula from the GM, they last because the GM doggedly perseveres in the face of adversity and keeps their game going, replacement players and all, by sheer force of will.
Sorry if that's not what you want to hear, but there is no formula. You do whatever your individual game needs to keep it going.