It says that flying monsters that are in the sky will move to the street that players are in. Do they only move when the move symbol matches the mythos card? Or do they automatticlly move to a street area where players are during the monster movement phase?

Except in the case of a gate burst on a seal (when all flying monster move), they will only move when their move symbol matches the mythos card.

ricedwlit said:

Actually, I think flying monsters move whenever a Gate Burst card is drawn, regardless of if it blows a seal.

I have never experienced a gate burst (probably because I don't have a mythos that has a gate burst), and I only 'swoop' a flyer if the symbol matches. It won us the game once, when the nightgaunts decided to go to sleep up there

Tibs said:

That's one of those things, that do Flyers possibly get a double-move if the GB also has the correct symbol? Flyer in a different town than an investigator on a street, GB drawn, Flyer pops to the Sky, then during the regular monster move part of the Mythos phase, get's a second move, drops down on the investigator.

Have another question about flying monsters. If they are also elusive, do they stay in the sky even if investigator is standing in street?

I don't believe any monster "double-moves." It either moves or it doesn't, similarly as how you're either delyed or you're not. If a Gate Burst shows a flying monster's movement symbol, then it's just redundant.

Tibs said:

Guess it goes to show that no matter how many times you read the rules, there's more to learn. I'd always played it as they move only when the seal is burst, but checking the rules (DH page 6) validates Tibs claim: they move whenever a gate burst is drawn.