To be honest, this probably requires a rules clarification before it can be considered a bug. I believe the functionality should work as I'll describe, but that may not be the case.
I created a Monk of Level 3, using a Variant Human that provided a feat at level 1 (the same would apply if I'd used another race and chose a feat at level 4). For the Feat I took Fighting Initiate, and selected Unarmed Fighting as a style. If the monk is carrying no weapons, that provides damage dice for unarmed strikes of 1d8 plus bonuses (which is higher than their martial arts dice at this level).
So, the fighting style description says my monk can now punch as an Attack action to do 1d8+STR. However, the end result is actually 1d8+DEX so the martial arts functionality of the Monk IS being used, as per "While you are unarmed or wielding only monk weapons and you aren’t wearing armor or wielding a shield, you can use DEX instead of STR for the attack and damage rolls". All is well.
Now we look at Flurry of blows. This is still showing an unarmed strike damage of 1d4+DEX. So it looks like the Unarmed Fighting fighting style is not being applied. I can understand that some may say this is how it is supposed to work, but the only evidence I can find supporting that, is it currently working the way it does (which as I say, could just be a bug or oversight in the logic). The description for Flurry of Blows reads "you can spend 1 ki point to make two unarmed strikes as a bonus action" - it doesn't explicitly mention martial arts, it just says Unarmed Strikes, and so I would absolutely expect the style gained through Fighting Initiate to apply here as well.
I expect that all of the above would be the same if I had multi-classed a Fighter/Monk instead of using a Feat, but I haven't tested that.
The attack added by Flurry of Blows is an Action hard coded to that feature and not coded as a regular Unarmed Strike. As such, the system doesn’t have the ability to apply the Fighting Style to that attack. It’s a system shortcoming.
"It’s a system shortcoming" - I figured that might be the case, thanks. I realise that DnDB is a complex beast, and I don't know where they draw the line between system shortcoming and bug. I guess I wait and see (and tbh, manually rolling the correct die is not a big pain).
They distinguish the difference as a Bug is something that is designed to work a certain way and doesn’t and a system limitation is something they haven’t been able to design that way yet.
To be honest, this probably requires a rules clarification before it can be considered a bug. I believe the functionality should work as I'll describe, but that may not be the case.
I created a Monk of Level 3, using a Variant Human that provided a feat at level 1 (the same would apply if I'd used another race and chose a feat at level 4). For the Feat I took Fighting Initiate, and selected Unarmed Fighting as a style. If the monk is carrying no weapons, that provides damage dice for unarmed strikes of 1d8 plus bonuses (which is higher than their martial arts dice at this level).
So, the fighting style description says my monk can now punch as an Attack action to do 1d8+STR. However, the end result is actually 1d8+DEX so the martial arts functionality of the Monk IS being used, as per "While you are unarmed or wielding only monk weapons and you aren’t wearing armor or wielding a shield, you can use DEX instead of STR for the attack and damage rolls". All is well.
Now we look at Flurry of blows. This is still showing an unarmed strike damage of 1d4+DEX. So it looks like the Unarmed Fighting fighting style is not being applied. I can understand that some may say this is how it is supposed to work, but the only evidence I can find supporting that, is it currently working the way it does (which as I say, could just be a bug or oversight in the logic). The description for Flurry of Blows reads "you can spend 1 ki point to make two unarmed strikes as a bonus action" - it doesn't explicitly mention martial arts, it just says Unarmed Strikes, and so I would absolutely expect the style gained through Fighting Initiate to apply here as well.
I expect that all of the above would be the same if I had multi-classed a Fighter/Monk instead of using a Feat, but I haven't tested that.
The character I made, if that is useful.
https://www.dndbeyond.com/characters/76940320
The attack added by Flurry of Blows is an Action hard coded to that feature and not coded as a regular Unarmed Strike. As such, the system doesn’t have the ability to apply the Fighting Style to that attack. It’s a system shortcoming.
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"It’s a system shortcoming" - I figured that might be the case, thanks. I realise that DnDB is a complex beast, and I don't know where they draw the line between system shortcoming and bug. I guess I wait and see (and tbh, manually rolling the correct die is not a big pain).
They distinguish the difference as a Bug is something that is designed to work a certain way and doesn’t and a system limitation is something they haven’t been able to design that way yet.
Creating Epic Boons on DDB
DDB Buyers' Guide
Hardcovers, DDB & You
Content Troubleshooting