Recently, I have audited a clinic with Alfredo Hernandez, Spanish born, Portugal trained dressage master with special talent for piaffe and passage. The clinic was very interactive and Mr, Hernandez welcomed questions from the audience. He also was very black and white with riders on what he wanted and he didn't back off until he got it. Below are the excises and ideas I picked.

Warm-up Leg-yield in walk, sometimes interrupted by either turn on the forehand or on the haunches, more advanced horses mixed a half-pass into it. "Trot - on the bit, through, otherwise do not trot" were his words, tempo control, he wanted slow tempo with engagement and swing. Think passage in posting trot. He got after riders for giving outside rein away! Almost every rider had to shorten their reins.
Canter warm up was specific to the work planned in canter. To collect canter more image of "holding the pee", "release the pee" for the medium :)

Leg-yield to turn on the forehand was popular exercise not only in walk but in trot too. The reason it helps the horses to come through and connect on outside rein. Also after the turn leg-yield was easy for a horse
Half-pass to turn on the forehand (flexion same direction as the turn) this one is for more advanced horses and can be done in all gaits. Specific quality of this exercise is the turn on the forehand done with the same aids as a half-pass. For example, half-pass left then keep the left bend and ask your horse to turn haunches more to the left from the right leg describing a circle around forehand (describes smaller circle). This version of the turn on the forehand is common in Europe but not on American continent. I have done it in Russia when I was learning how to ride. It considered the final version of the turn. The typical one here was considered then only a stage in the young horse training. Of course, in canter the turn is very big and front legs describe quite a circle. I was able to do it on Santo with 8-10 m diameter. This is how I have done it: canter half-pass left from quarter line to quarter line, start half turn on the forehand to end up on the first quarter line again and then finish with left half-pass to the wall. Second half-pass feels so easy after the turn LOL
Variation Walk/Canter pirouette out of the turn on the forehand

10 m circle to half-pass in trot. The challenge is the quality of trot - slow, swinging, very cadenced trot. He made a rider to repeat this exercise several times until he was satisfied with the results.

Piaffe/Passage work on piaffe in hand with long whip - very collected walk steps to very short active half steps. He was quite sharp with the whip if horse showed sluggishness in response and he didn't let the horse run past him. Working the horse between these two alternatives he showed amazing feel and quickness to praise the horse for correct steps. Exercise: Sharp tap on the belly with bamboo stick at the halt, collected trot after (tap for the lift in the back)
Introduction to passage happened with a thick bamboo stick. As I understand Joseph Newcomb learned his bamboo technique from Mr. Hernandez. Tapping the front legs in the rhythm of the walk steps caused the horse elevated steps in front. Mr. Hernandez quickened the tempo of his taps to cause the horse to start trotting. Bamboo encouraged elevation and lift which created slower tempo and higher steps more resembling passage then trot. Rider was quiet and not interfering unless directed by Mr. Hernandez. Exercise: 20 m circle in collected trot, ask for renvers, straighten and ask for passage. He wanted the rider to carry a bamboo stick to tap the shoulder.
Piaffe improves canter. Passage can improve piaffe and vise versa. The horse dictates how you start teaching these movements, either one can be first or both together.

Part II will be dedicated to work on pirouettes.

Happy riding...