Specifically for PBP, I couldn't make the DH initiative work in a reasonable timeframe (reasonable to me being everyone goes at least once within 2 days). Like, throwing it to a specific player according to how often they tend to check? That sort-of works when the most active player checks in 3 times a day, then you pass it to the girl who checks twice a day, then juggle the last two players who check once per day, but we just ended up missing lots of time that way.Oh boy, yeah, good point. Maybe doing as Derik suggests and when the spotlight goes to the players, the gm throws it to a specific player, asking how they respond to the gm move.
Using an initiative tracker is faster, but can also take a very long time if it's not the players turn when one of your Once A Day checkers swings by and see's they can't post. You could set up some kind of delayed post situation, where they write their action but it doesn't happen until a player turn comes up again. There's probably some merit in that.
In the end, we went with Sided Initiative instead. All of the players take their actions and make their rolls, then I count up how many Failures or Fears they rolled and arbitrate the results + take the appropriate amount of GM moves. And you make a good faith effort to not dogpile anyone specifically with your glut of actions.
That's certainly not playing Daggerheart as intended, but the alternative took catastrophically longer. DH initiative might work okay in PBP with an extremely responsive group where everyone checks multiple times a day, though.