If it's only moving when you slide from screen to screen, then that's not a live wallpaper. A live wallpaper moves constantly, even if you're not moving from screen to screen, and depending on the phone, the lock screen too. Some let you interact with it.
And yes, a live wallpaper does drain your battery. The best wallpaper, of course, is one with no animations. A plain color is another solution to saving battery power.