Does changing ability power halfway through an ability change it's damage?

Hey guys, I've been tampering with different builds recently, specifically speed ones with celerity as mobility is extremely useful in this game. In doing so I noticed some champions in combinations with items can get extreme bursts of speed which equates to a massive temporary increase of ap or ad. This made me wonder something quite important; when casting abilities does temporary change in the ap or ad scaling work only when the ability is first used or when it does damage? Some examples would be vlads ult or ashes ult, the first where the ability "hits" immediately but doesn't deal damage till after a set delay, or the second where the projectile is still in transit until it hits a champion. Another would be malzahar's e, does the DoT change with the changing scaling? This is super important as you may find yourself with a load of extra scaling at the start of the cast, only to find it wears off and deals reduced damage by the time the damage is dealt, or half way through only to find it doesn't change and you missed an opportunity.

Really curious so I would appreciate any examinations!