Elemental Affinity & Empowered Evocation -> DoT Spells

[RulesJD]

Assuming correct setup, how does EE and EA interact with duration damage spells?

For example, Wall of Fire. Does the (idiotic) current rule establish that you only add your casting mod to the initial roll for when the spell is cast, and no subsequent damage rolls (a monster tries walking through the wall, etc)?

Or is it like Magic Missile, where you roll the damage when you initially cast the spell, add your casting mod, and then use that damage roll each time it's needed, thus effectively making it apply each time an enemy is damaged?

I can see arguments both ways, but can't seem to find much guidance. This impacts a ton of other spells (Storm Sphere, Elemental Bane, etc) and quite frankly any ruling that lets a cantrip (Eldritch Blast or GFB with 2 targets) out damage a 2nd level spell like scorching ray is clearly wrong, but c'est la vie.

[Spectre9000]

Assuming correct setup, how does EE and EA interact with duration damage spells?

For example, Wall of Fire. Does the (idiotic) current rule establish that you only add your casting mod to the initial roll for when the spell is cast, and no subsequent damage rolls (a monster tries walking through the wall, etc)?

Or is it like Magic Missile, where you roll the damage when you initially cast the spell, add your casting mod, and then use that damage roll each time it's needed, thus effectively making it apply each time an enemy is damaged?

I can see arguments both ways, but can't seem to find much guidance. This impacts a ton of other spells (Storm Sphere, Elemental Bane, etc) and quite frankly any ruling that lets a cantrip (Eldritch Blast or GFB with 2 targets) out damage a 2nd level spell like scorching ray is clearly wrong, but c'est la vie.

By Errata, Elemental Affinity applies damage only once over the life of the spell. I've yet to hear someone say they agree with this ruling. Talk with your DM, and he'll probably overrule Errata, which in this case, is really the only sensible thing to do. Elemental Affinity via Errata, applies a -5 to +5 bonus to a spells damage period, which is so underwhelming against comparable class features such as immunity to charm, spell resistance, bonus to all saving throws, damage reduction, spell slot regen, Con saving throw proficiency, etc.

[RulesJD]

Technically speaking, I don't think it says anywhere that you must re-roll the initial damage for non-hit spells. For example, you need to roll to hit for each ray of a Scorching Ray, which necessitates rerolling damage. However, for DoT spells like Wall of Fire, it just says they take the damage at certain times or save for half. So roll the damage once, apply your modifier, use that damage roll each time it comes up, and you get around the errata.

I know people aren't used to DoT spells working that way, but is there any rule that says they don't?

[Citan]

Technically speaking, I don't think it says anywhere that you must re-roll the initial damage for non-hit spells. For example, you need to roll to hit for each ray of a Scorching Ray, which necessitates rerolling damage. However, for DoT spells like Wall of Fire, it just says they take the damage at certain times or save for half. So roll the damage once, apply your modifier, use that damage roll each time it comes up, and you get around the errata.

I know people aren't used to DoT spells working that way, but is there any rule that says they don't?

I'd say it depends on how people envision how the spell and its interaction with creatures works.
To support yours, you could say that since it's a magical effect, the strength of the flames (to stick with your example) is constant and fully "hit" anyone that fulfill the conditions.

To support the other, you could say that since a creature makes a saving throw, it means (whether it fails or succeed) that it tried to do something to avoid damage, hence introducing a variable hence needing to reroll damage.

Personnally, as a DM, I'd use the first variant if I have a good reason NOT to overrule the "apply feature once only" ruling.
So far, I see none so I ignore it.

With that said, another reason to go this way (roll spell damage once) is for the sake of tracking simplicity, both for DM and players. :)
But only if everyone agrees with it (because I'd have to apply this method to everyone, and people may not like the idea to risk being stuck for a whole spell's duration with a ****ty damage roll).