Apocalypse World, Play By Post?

If you're playing on a forum or play-by-post, how do you use 'at the start of the session...'-style moves?

I really don't recommend this style of play, but if you must, you can do it by calms in the action. Think of it like deciding chapters in a book. You can also do it after each participant has made a certain number of moves + the next pause.

Yeah, I think the idea would be to attach that trigger ("at the start of the session") to something else that you do in online play.

Perhaps, "When you skip forward to a future time, days, weeks, or months later..."

(By the way, I know some people who have run some amazing PbP AW games, although I've never participated. So it's definitely doable!)

Just do it by feel. In my experience, asynchronous online play often benefits from specific resets/breaks already -- activity tends to follow a surge/lull trend, and declaring a 'new session' is a good way to ensure that the lulls do not result in the inactivity-death of the game.