A proper smooth transition that does not change the cadence of the foot fall is done when the horse initiates the transition with its hind end.
There are 2 types of poor transitions the ones that you describe and also the transition where the horse actually stops then starts again within the same stride.
So why is your horse speeding up? I am going to guess that he is falling forward to the transition. Think of a Baby when they first learn to walk if they lean to far forward they have to scramble to keep their balance. Your horse is scrambling to keep the same ground speed at the new gait.
To do a proper transition the horse has to shift a little weight to its hind end since the hind end is the engine we need to make sure it is in control of the transition.
There are 2 ways to do this depending on what you end goal is for the horse, one and the most common for English horses is we elevate the shoulders and have the horse round at the loin. The most common way to shift weight back for the western horse is to have the back round under the saddle with the loins and whithers both softening, this in effect allows the hind legs to drive slightly deeper under your seat and changes the horses balance point. ( I wrote a blog post on the difference between round and elevated on my coaching blog that you can refer to for a more detailed explanation if you like)
So this is why transitions are used as one of the first steps in teaching our horses to shift wight to their hind ends because it does not take much of a shift but they have to respond to our cues in the same way to do a proper smooth transition as they do to perform collection etc. when they get further along in their training.
So How do you start to teach proper transitions? you need to develop your horses ability to go forward into contact and you need to develop your rhythm.
The above 2 descriptions of poor transitions can be taught to a horse by poor training or they can be natural for the horse, the horse that scrambles during transitions is usually pushy on our hands when we use them to guide as well, And the horse that stops then starts is usually very light mouthed and is always trying to short stride or stop when we use our hands to guide. Neither have been taught to accept the riders hands or legs, so that is the first place I would start in teaching proper transitions.
I am going to include a couple video's here that I think may help you understand what i am talking about in acceptance and also how to develop some acceptance in your horse.
These video's are only showing the trot but they can be used at the lope as well, the one on transitions with rhythm can be done at a lope as long as you can find your horses lope rhythm and to do that just relax your whole body and let the horse carry you and feel how your body wants to move when your horse is loping nice and relaxed. if you cant feel your horses rhythm one of 2 things is happening your horse isn't relaxed and using a natural non rushed cadence to it's stride or you are not relaxed. :o)
Hope this helps and if you would like to watch more video's I have made for Performance Horse Development you can visit my coaching blog and request an invitation to my subscription Blog. It is free until I get my new video's finished
To teach our horse acceptance we have to be effective with our hands and legs.. Rod Miller IPHDA & NRHA Professional