Any touch/release MUST be TIMED to the horse's ability to move because of it. I often see people touch/kick/spur double or triple time, but all that does is tense the belly...the horse cannot react/hop on that spur action.
Yes, as per Puck's good explanation, too. I liked what you, equitate, said about raising the rein, too.
When I was taking dressage lessons, I was told the way to freshen and strengthen the walk was by apply ing the correctly timed leg , alternating sides, to get a bigger, more step-under walk. It works, however, one can fall into the trap of doing it endlessly and then the hrose starts to require this just to maintain a good walk. Then, the beginner (as I am) can end up doing way too much contorting of the body to get that side this, side that squeeze and it all goes down hill.
With the mention of raising the rein, you bring to mind more of what I have been taught in recent years, where I pretty much just trail ride, and that is to "bring up the energy" in your body, and that alone will often be enough to increase the energy in the horse.j
I try to do that, then if no response , and I mean the response should be pretty darn immediate, I will tap the horse's sides with the inside of my ankle bone, and if that doesn't get a smart response I snap the whip against my foot. He leaps off!.
But, I would never cue every step in a walk or a forward trot. You might cue every step in a pirhoette, but not just walking down the arean.