Presumably you've taught your horse some of the basic whip aids...such as pointing the whip at the hock or hip for an upward transition from halt to walk in this case.
The other key to transitions is the voice aid. You ONLY give the voice aid for a transition 'when the horse is about to give it to you'. In other words, if the horse is cantering around and you want it to trot, you don't keep repeating 'trot, trot, trot, trot', while the horse canters round and round. Same deal, you don't let the horse walk around and around and around while you're saying, 'trot, trot, trot'.
When giving voice aids you elongate the words. On upward transitions you speak in a higher octave and raise your voice on the last of the word. On downward transitions you speak in a lower octave and drop your voice on the last of the words. You also change how the words are elongated.
Up transitions it would be 'ttttttrrrrrrrrrrrrrrrrrrrr - OT, or caaaaaaannnnnn - TER!
Down transition would be 'tttttrrrr-oooooooooooooooootttttttttt, or wwwwwwaaa-aaaaaaaaaaaallllllllllllllllllllllllllllllk
Ask the horse to walk, then 'push' the walk a bit with your body, raise the longeing whip and point it at the hock. If the horse doesn't begin to think trot, then you'll need to increase your body push and raise the whip to the point of hip. If still the horse isn't thinking about trotting, then you'll need to touch the top of the hindquarter with the whip. The moment the horse 'thinks' trot, you need to add the voice aid with lots of 'happiness' and inflexion in your voice.