How have I not talked about running combat yet?!
PSA for those of you interested; the Pathfinder 2 Playtest is officially available, and can be downloaded from Paizo.com for those interested. Interested parties looking either for a group or for a GM should either check in at Rondak’s Portal forums, or at Rondak’s Satellite; our Discord server.
Okay. Combat. Read on past the cut, folks!

Handling combat in play by post games is one of those tricky things that usually can be counted on to slow things down. How much depends very much on the GM involved; some prefer to ask each player for their actions in order, others want actions sent to them, others just have everybody post them publicly.
These methods and several more have things going for them. Which ones work best is generally up to the GM; consistent practices usually include time limits or deadlines, and generally involve the GM doing most of the rolling, or having players roll not only to hit, but also damage and anything else that might happen on their turn.
So, the basic methods I have seen or heard of, in no particular order are:
Free-for-all: The GM announces the start of combat. No initiative is used; actions happen more or less in order they are posted, with the monsters either acting first (surprise) or last (no surprise), and the GM summarising each round once all actions are in.
I’ve never seen this done, to be honest. It might work very well, with the GM rolling everything, whether posts are made publicly or privately to the GM only.
Strict Initiative Order: The GM calls for initiative, or announces start of combat and rolls initiative, and asks for each player’s actions (and possibly applicable rolls) in turn, then summarises the round.
This has the probability of being very slow, unless run in real-time chat, ie; Discord, Portal chat or Hangouts. This is also the most common method I have seen used, and is slightly less prone to players acting on old information; everybody knows the moment the monster goes down, for example.
Rough Initiative Order: The GM calls for initiative, or calls start of combat and rolls initiatives and takes actions (and applicable rolls) in batches; all before monster, all after monster, and summarises the full round.
This can be very efficient, more so if the GM and players are very familiar with each other. It can also be pretty slow, and it is easy to get mixed up where exactly a group is in combat.
Most of these will be more efficient if the GM does most of the rolling necessary; no waiting for a slow player which does slow things down.
If my fellow GMs know any other ways to run combats that I haven’t mentioned, please, feel free to mention them, here in the comments, on Discord or even in the Portal forums.