It's important to remember that despite common misconception, the horse's head carriage is merely a symptom of being on the bit and it has actually very little to do with head carriage. It's one of my least favourite terms out there.
When a horse is on the bit, it's whole body is working. It's engaged and, travelling with self carriage. This means that the horse is drawing its power and impulsion through the haunches, up along the back, and through the hands. When the horse does this, it seeks contact through the reins. Upper levels of dressage are now trying to include long rein work more and more into their tests, because it is a test of self carriage. Is the horse truely displaying self carriage, or is the rider just making it seem that way? If it doesn't really have self carriage, if the head is in but the back is hollow, if the hindlegs are not engaged, then the horse will just fall apart.
In fact, hollow back and disengaged legs, but head in is a classic sign of what "on the bit" leads people to believe vs what it actually means. Try it yourself. Get down on your hands and knees, and hollow your back. Your head will naturally rise and will be difficult to put down. If you bring your legs under you a bit and hunch your back slightly, your head will drop. Now imagine putting weight (a rider) on that, and see which is preferrable. Another common issue with a horse incorrectly being "on the bit" is it's often accompanied by rollkur and/or a broken neckline.
So how is this properly achieved?
By using the Scales of Training as a base. Obviously every horse is different. Getting a horse on the bit is not something to be achieved in a single session. It may even be months or years before a horse is athletic, supple and responsive enough to achieve it. Lungeing, polework, backing up, and lateral work are all exercises that can be used to help achieve the first few stages in the scales. You can't really have true self carriage or collection without the other steps. Your horse should be working with you, not against you or because you make it. A contact is part of the scales of training, and therefore part of being on the bit, but not the main part. Done correctly, it's a whole body work out for horse and rider.
To get your horse on the bit, you have to use your legs. The movement needs to go from the back, your horse has to be balanced, to have his hind legs under him. Once this part done, you have to have a smooth and constant contact with your hands, so that your horse will be able to get on the bit, he'll trust your hand.
You can do a lot of transitions, halts, figure-eights ... while always looking for energy.