After crossing the wire when racing, most horses are tired, so that does not equate with dropping the reins to slow them down. It usually is a cue, but not every horse will respond to that. There are plenty of horses that will continue to want to go forward, and hopefully the jock is strong enough to pull them up.
As far as morning training, you will have your tough horses, no matter what. Some will relax when you loosen the reins, and some will run off. Same with lengthening your stirrups, works for some, others will leave you yelling for an outrider.
Dropping the reins on some horses will not be enough to pull them up, sometimes (and often) it will really take some brute strength to slow these guys down.
Most racehorses are schooled as babies to take hold of the bridle. They learn to lean against a riders hands, the thought is to make them work a little harder rather than galloping around the track with their heads in the air. As they mature, some learn to gallop in a relaxed hold, while being fit enough to keep their fitness per say.
What you will see when watching a race is typically a horse early on will run with a longer rein, then the jock takes up the slack to push on the horse…so hands will change position from the withers to higher on the horses neck, cueing the horse and urging him forward.
To try and answer the question about pulling up a horse who’s injured, it does take strength, and if you watch tapes of races when a horse breaks down you will see the jock taking a hold and pulling back for all he or she is worth. Dropping the reins at this point would be counter productive as the horse will probably lower their heads and fall. There is really no “trick”…there may be some see sawing going on, but no different than trying to get a runaway horse to slow down. Of course the one rein stop is not going to work on a horse that has an injury, recipe for disaster.