As you've seen across the various iterations of Fate, boosts don't have a set duration but they're "super-transient"1 because they represent fleeting advantages and disadvantages. The rule of thumb is that a boost should be kept around until there's a good opportunity to use it--and not much longer.
Boosts... go away on their own fairly quickly—usually after the next action when you could use them... (from A Clarification about Boosts)
Boosts aren't aspects and don't even really need names; we won't spend a lot of effort tracking them over time. While wiping away a full aspect usually takes an Overcome or Create Advantage action, boosts melt much more easily.
So use boosts as soon as you can because after you've turned away a chance to use a boost, you've forfeited its narrative protection: as the scene moves forward, the boost gets left behind.
In my experience, this usually means a boost lasts through the player's next turn but rarely through a full second round after that. I've also known groups who don't write boosts down; instead, their boosts last until forgotten in the excitement of the scene. I think of boosts as free invokes without aspects to ground them.
1 While Fate Core describes boosts as a "kind of aspect," that's an observation of resemblance rather than a mechanical equivalence. Boosts have aspect-like impact on the narrative flow, and get named in an aspect-like style, but they don't interact with the other mechanics in aspect-like ways--eg, they don't touch the fate point economy through invokes or compels.
