Laura Graves gives advice and tips on how to improve connection while riding. From the article:
Your horse is responsible for creating consistent energy, whether he’s walking, trotting, cantering, piaffing or doing an extended trot. This consistent energy shouldn’t be mistaken for “impulsion.” True impulsion requires a certain amount of strength that develops over time. The horse must simply, on his own, be self-motivated, creating steady, dependable forwardness so the same amount of energy gets to the bit every single stride. It’s the rider’s job to hold the horse accountable. Young horses might be timid or perhaps they pull (they are all different), but teaching them to create their own consistent energy and be sensitive to the aids will also teach them to accept the bit—as opposed to struggle against it.
When the horse doesn’t maintain a steady forward energy, he may momentarily go faster when the rider puts her legs on, but then slow down, requiring the rider to use the legs again. Some horses have the reverse tendency. They go faster and faster. Those horses need to respect the half halt. In any case, you don’t want to be using your aids to constantly regulate the horse’s speed. Horses that are consistent find their jobs more enjoyable. The ride is just nicer when the flow of energy stays the same. Then that energy can be recycled by half halts, making the work easier.