Animating with depth is slightly harder than animating side movement in 2D indeed.
But basically Spine gives us the tools to animate characters in all directions. For many cases, it is sufficient to scale bones or meshes along their main axis, just play around with it. By adding gradient shadow layers with fading opacity during animations, the effect of depth can be emphasized.
In animations like wielding a sword or aiming with a gun, you can swap the images that completely change their face.
A typical walk cycle towards the camera involves scaling of legs, (smooth & gradient) shadows and swapping of the feet:
http://media.tumblr.com/6122246b597b619b3ff7eb96ac567264/tumblr_inline_mjkkqgJkPe1qz4rgp.jpg
I think nearly every animation can be split up into its simple components that change (rotate, scale, sheer, shadow, swap). Anyhow, it’s still a challenge to do and will need some practice in Spine.