There are multiple player races that can fly, including the Aarakocra (comes with a 50' fly speed).
Are there any official rules on how the flight of these races is affected by difficult weather conditions, such as a storm or extreme winds?
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The Wilderness Survival section in the Chapter 5 of the DMG presents rules for strong wind conditions (pg. 110). It is as follows, emphasis mine:
A strong wind imposes disadvantage on ranged weapon attack rolls and Wisdom (Perception) checks that rely on hearing. A strong wind also extinguishes open flames, disperses fog, and makes flying by nonmagical means nearly impossible. A flying creature in a strong wind must land at the end of its turn or fall.
The rules of heavy precipation in this chapter don't present any guidance regarding flying creatures, but you could make the "flying area" as difficult terrain, effectively halving the flying speed of a creature.
Various weather related spells, such as control weather, control winds, gust of wind, and wind wall can be used to hinder/difficult the flight capabilities of a creature.
One option is that the DM might determine the storm to constitute difficult terrain; then the fly speed would be halved.
You move at half speed in difficult terrain--moving 1 foot in difficult terrain costs 2 feet of speed--so you can cover only half the normal distance in a minute, an hour, or a day.