There are mechanical reasons in the 5e system why actual changes in initiative order once it has been established do not exist and are not permitted. There are ways that it can be abused and broken unfairly, due to the way other skills and abilities are designed to work.
If you want one quick example, take this one: [...]
Monk's tturn would be next - DM says "Actually, the Monk chooses to delay her turn until later in the initiative order. She's going to go after her minions now."
- Players, who should have recovered by now, are *Still Stunned* - because the monk has not taken her turn, and the monk having her turn is the timing for when the stun ends.
- Minions get a second turn to attack the stunned players for free.
The villian has just gotten a second full turn out of one application of stunning strike, without having to spend any additional resources at all, and has saved five ki points and risk of saves... all for no cost, just because she 'decided' to move herself further down the turn order.
I mean, yes this is technically true according to RAW (IF delay happened to also be RAW without any other rule changes). But I feel like it's incredibly intuitive to say that "No, the players become unstunned at the end of the initiative spot where the monk *would* have gone. All tables I know who allow the Delay action run it this way.
It obviously adds more bookkeeping to keep track of original initiative spots for multi-turn effects, but imo that's worth the flexibility and fun that having a Delay option adds. I always allow and advocate for a delay option in games I'm in.
If you use a Delay rule, you're already adding homebrew so might as well add an additional homebrew caveat to that rule saying "All multi-turn effects remain at their original initiative counts; no bypassing or lengthening multi-turn effects through Delays."
