Draconic Bloodline sorcerers gain the Elemental Affinity feature at level 6:
Starting at 6th level, when you cast a spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell. At the same time, you can spend 1 sorcery point to gain resistance to that damage type for 1 hour.
The description of the booming blade spell (SCAG p. 142) states:
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails. On a hit, the target suffers the attack’s normal effects, and it becomes sheathed in booming energy until the start of your next turn. If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends.
This spell’s damage increases when you reach higher levels. At 5th level, the melee attack deals an extra 1d8 thunder damage to the target, and the damage the target takes for moving increases to 2d8. Both damage rolls increase by 1d8 at 11th level and 17th level.
I realize those two can't work together normally since none of the draconic ancestry options are associated with the thunder damage type. However, let's assume we have a dragon ancestor option that lets Elemental Affinity apply to thunder damage, or a spell that works like booming blade but deals the damage type associated with one's Dragon Ancestor, with the same wording.
How would this hypothetical scenario work out? Above level 5 it's pretty obvious that you deal your bonus damage on impact, but below that, your initial cast only deals weapon damage - but is there anything RAW preventing you from dealing your Elemental Affinity damage on the move damage, since it's the first time this spell deals your chosen element?
(Since Elemental Affinity only applies once the character is 6th level, assume for the sake of argument that we're talking about an equivalent homebrew ability gained at an earlier level.)