This seems a bit similar to the initiative system for Shadowrun.
In that system, initiative was rolled.
Then everyone acted in turn order of what they rolled.
Then you subtract 10 from everyone's initiative.
Everyone who still has a number above 0 gets to act again in new count order.
Repeat this until no one has anymore actions.
Roll initiative again, getting new counts.
Repeat the process until combat is over.
This type of system seems like a good compromise between only rolling initiative once and rolling initiative every round. It also has a chance to change the combat significantly from one initiative roll to the next because you may get a good roll the first time, but not so good the next time.
Of course that system had a bunch of effects and dice pools that refreshed when initiative was rolled, so there were a lot of things built into that system. As far as 5e goes I can think of only a few things that happen upon initiative roll, like Battle Master refresh and Monk 20th level ability, and several other things like Assassin abilities that could get really overpowered or underpowered depending on rolls and builds.
In that system, initiative was rolled.
Then everyone acted in turn order of what they rolled.
Then you subtract 10 from everyone's initiative.
Everyone who still has a number above 0 gets to act again in new count order.
Repeat this until no one has anymore actions.
Roll initiative again, getting new counts.
Repeat the process until combat is over.
This type of system seems like a good compromise between only rolling initiative once and rolling initiative every round. It also has a chance to change the combat significantly from one initiative roll to the next because you may get a good roll the first time, but not so good the next time.
Of course that system had a bunch of effects and dice pools that refreshed when initiative was rolled, so there were a lot of things built into that system. As far as 5e goes I can think of only a few things that happen upon initiative roll, like Battle Master refresh and Monk 20th level ability, and several other things like Assassin abilities that could get really overpowered or underpowered depending on rolls and builds.