It's a somewhat common thread. It's in the body's nature to not release the bladder when in an "unfamiliar" situation. The brain puts the brakes on it unless you really intend to pee. For most, this can be a very difficult thing to overcome in unusual situations. While clothed, laying down, sitting, walking, riding in a moving vehicle, in any position in public, even just trying to use a public restroom, it can be a struggle.
Walking is especially difficult because not only are you IN an unusual circumstance to be wetting, but your body is physically busy doing something else, that it NEVER normally multitasks with peeing. So that just makes it really difficult to concentrate. Going up or down stairs adds to the difficulty, as does running. As anyone will tell you, it just takes time and practice, to make gradual progress.
While I can sometimes just "let it go" when the bell rings, I still frequently need to concentate to some degree or other, especially when walking or riding in a car. But I'm "training" my way out of an extremely "shy bladder" aka "stage fright" - where I was unable to start going in any public restroom. It was only recently that I was able to use a urinal while someone else was using another urinal in the room. (though now that I'm diapered 24/7, using a urinal is less of a concern...

)
As for the leaking... most diapers have "standing leak guards", which most people will just refer to as "leak guards". But they're
standing for a reason. They're designed to keep the pee IN your diaper while standing, instead of rolling out the legs left and right, long enough for the material to absorb and wick it up into the padding in the front and back. If you're leaking while walking, you're either not wearing a diaper with leak guards, have a diaper thats already too wet, are flooding way too fast for the diaper to keep up, have a diaper with crappy short leak guards, or have a problem with your taping method.