Do you want the long or short version??!!!
Half halting is something that develops over time, much like collection.
You start by teaching basic transitions. Until you can make a downwards transition as soon as you THINK downward, then you won't be able to ride a half halt.
When you transitions are on the spot, then start riding towards a downwards transition, and just before the horse changes gait, ride forward again.
This gradually becomes so refined that there is a nearly imperceptible shift of balance to the hind legs, with seemingly no aid from the rider.
A true half halt is ridden through only the seat, with no rein required, and is intended to rebalance the weight over the hind legs, and to activate the hind legs while bringing the wither up.
It is something that takes years to develop, there are whole books written on the subject of half halts, with much debate and ideas on how to ride a half halt - whether to ride it with the lower seat, to half halt through the base of your neck, to sit deeper or to lighten the seat... etc.
To teach the half halt to a horse you must understand it. Here is what I have posted in the past:
-------------------------- Half Halts ------------------------------- Instead of going for a harsher bit (which will just make his mouth harder - then it'll stop working) instead work on the half halts.
Start by trotting and asking for a halt. Rider accomplished this by: doing a stomach crunch (core), stopping your hips, planting elbows on your waist and pushing down (NOT forward) with BOTH stirrups. I like to exhale at the same time which, besides being a verbal cue for the horse, also makes your body "heavier" since you no longer have as much air in your lungs. If that doesn't work add in a ‘whoa’ - object is trot to halt with ZERO walk steps and hind legs together (NOT one in front of the other).
Once that's good make every other transition a trot Half Halt (HH) where at the last second as horse is stepping underneath itself with its rear legs the rider allows their hips to move and the elbows to come forward about 1/2 an inch (think of it as softening your elbows). Horse should continue trotting after rocking back on its butt during the HH. If that doesn't work go back to full halts and work on tour timing - it MUST be correct - horse MUST be stepping underneath itself with rear legs to lift front end and stop falling on its forehand (which is what is happening now).
Now once that's good start the ‘canter to walk’ (or ‘canter to halt’) work using the same principals.