Breaking up your Movement doesn't take separate actions or triggers.
Consider that you can do the following in 1 turn:
Move 15 ft, attack a target (action), move 5 ft, attack another target (extra attack or bonus action off hand), move 10 more feet.
So if you're playing a flying character, and the DM has house ruled a 1000 ft/round drop, the DM has increased your movement by a potential 1000 ft in a downwards direction for free.
Falling while flying doesn't cost you an action. All you have to do is stop flying. Starting flying again is also movement, and does not cost you an action. If I was your DM, your flying speed would be how far you could move in any combination of directions, regardless of whether you were going up, down, left or right. Your DM apparently thinks gravity is sympathetic to the atmospheric pressure of Venus, and has multiplied the effects by 30.
In actual physics:
Distance = 0.5 (gt^2); where g = gravity @ 9.8 meters per second, and t = time in seconds. End result = 176.5 meters, or approximately 574 feet
But the bottom line is this:
Movement doesn't take an action.
So long as you haven't moved your speed in your turn (and the DM ruling of falling 1000 ft/round is not you moving your speed, that's him moving you with his house rule), you are free to move it, by RAW, in whatever order you want.
Ref: PHB pg. 190
Breaking up your move
You can break up your movement on your turn, using
some of your speed before and after your action. For
example, if you have a speed of 30 feet, you can move
10 feet, take your action, and then move 20 feet.
So fly, stop, fall however far you want, then fly the last bit of your movement. You control your movement, and if he says you can't just choose to drop, tell him that's fine, but if you get knocked prone in the air then the enemy can't just choose to drop you 1000 ft. Frankly, it sounds like your DM hates flying combat and is trying to get you to play a grounded character. You might want to consider discussing that with him/her.
Xanathar's has been released since this answer was submitted, and there is a straightforward 500ft/turn rule now.