So if a creature with flyby attack hits and snatches a creature, do they get to finish their movement? Also, how would grappling affect the hover check (since the grappling creature can't make a normal movement to keep flying)?
Unless you take the -20 to avoid the grappled condition the full rules of that condition apply, including the inability to move. So no further movement until the next round when you make a grapple check to move the grapple.
Hover would be fine since you use it to keep from moving while flying, but you need to actually make the check every round.
Flyby Attack (Monster)
This creature can make an attack before and after it moves while flying.
Prerequisite: Fly speed.
Benefit: When flying, the creature can take a move action and another standard action at any point during the move. The creature cannot take a second move action during a round when it makes a flyby attack.
Normal: Without this feat, the creature takes a standard action either before or after its move.
Grappling
As a standard action, you can attempt to grapple a foe, hindering his combat options. If you do not have Improved Grapple, grab, or a similar ability, attempting to grapple a foe provokes an attack of opportunity from the target of your maneuver. Humanoid creatures without two free hands attempting to grapple a foe take a –4 penalty on the combat maneuver roll. If successful, both you and the target gain the grappled condition. If you successfully grapple a creature that is not adjacent to you, move that creature to an adjacent open space (if no space is available, your grapple fails). Although both creatures have the grappled condition, you can, as the creature that initiated the grapple, release the grapple as a free action, removing the condition from both you and the target. If you do not release the grapple, you must continue to make a check each round, as a standard action, to maintain the hold. If your target does not break the grapple, you get a +5 circumstance bonus on grapple checks made against the same target in subsequent rounds. Once you are grappling an opponent, a successful check allows you to continue grappling the foe, and also allows you to perform one of the following actions (as part of the standard action spent to maintain the grapple).
Move
You can move both yourself and your target up to half your speed. At the end of your movement, you can place your target in any square adjacent to you. If you attempt to place your foe in a hazardous location, such as in a wall of fire or over a pit, the target receives a free attempt to break your grapple with a +4 bonus.
A Flyby Attack is not meant to help facilitate a move-grapple-move, per RAI. It's meant to give an attack in the middle of a move action. However, RAW, it says you can perform a Standard Action in the middle of a move action. And Grappling is a Standard Action, so...
Also of note, Snatch is a free grapple attempt on an attack. So if you DID perform an attack in the middle of a Flyby Attack, you can automatically attempt a grapple for a free action...
However, the Grappling rules say that you cannot Move a target you're grappling until round 2; you have to make a grapple check to start the grapple in round 1, and then round 2 you make another grapple check to Move them.
RAW, I'd say no.
RAI, I'd also say no. Only because I don’t think Snatch was ever supposed to work with Flyby attack.
Honestly this makes no sense though, Eagles, hawks, falcons, owls, and every other predatory bird under the sun make Move-Grapple(Snatch)-Move actions IRL on a daily basis, so w/e. I see no reason why a dragon or anything else couldn’t do the same thing, especially since Snatch is used on targets three size categories lower...
Since no feat exists to my knowledge that would allow this to be 100% kosher RAW, for Rule of Cool, I'd say sure, but you can only move half speed after the grapple.
Thanks for the feedback!