I use a lunge line to encourage her. Hold it in your left hand, so that way it's on the outside and not between you and your horse. For showmanship you get penalties for jogging before your horse does, so I turn up my energy, walk faster, click, and give a "whip" with the lunge line, however, DO NOT touch the horse with it. Keep on asking till he trots. Once he does, jog for a few strides then praise him.
When you lean forward slightly, your horse should go with you. To train, you can use your lead rope to swing it behind your back and tap (don't whip) the horse for a little extra encouragement to go forward, until your horse starts to understand your body cues.
Give a slight tug then start to kinda run once he trots start actually run. Then to whoa pull back slightly; Don't touch the horse !