...the solid fog is so thick that any creature attempting to move through it progresses at a speed of 5 feet...
So that a flying creature would have to reduce its flying speed to 5 feet when it is in a solid fog.
And if the flying creature does not have better-than-average maneuverability, then it will fall.
If a flying creature fails to maintain its minimum forward speed, it must land at the end of its movement. If it is too high above the ground to land, it falls straight down, descending 150 feet in the first round of falling.
Interestingly, it seems solid fog does not stop a creature from dropping or limit its dropping speed to 5 feet. Instead, it says:
A creature or object that falls into solid fog is slowed, so that each 10 feet of vapor that it passes through reduces falling damage by 1d6.
(If it simply reduces the speed to 5 feet, then there won't be any falling damage. As a comparison, Featherfall reduces the falling speed to 60 feet which has no falling damage.)
So I assume that if a flying creature is trapped in a Solid Fog, it will trigger falling rule since 5 feet is less than its minimum forward speed, and hence instantly leave the Solid Fog area and either take falling damage or can start flying again from next round.
Is this the correct way to read these rules?