Everything works fine in the editor but some weird stuff happens with the rendering when I build for my target device.

Strange things sometimes happen with Spine-Unity's bundled default shaders when you build for certain target devices, most commonly with the lit shader.
Sometimes causing weird artifacts or compilation errors.

The Skeleton shaders are actually mostly fixed-function shaderlab shaders so the problem is usually on Unity's side.
We usually report these things to Unity but it takes weeks for them to get back to us, but often, they're bugs that get fixed in subsequent patch releases of Unity itself.

Among the problems we've reported to Unity were related to Xbox One, tvOS and XiaoMi RedMi.
If you notice any problems regarding shaders, please do let us know.

Make sure your problem isn't just a basic Premultiply Alpha problem though! That's easily solved with the correct export settings.

For more chances that something works where the default shaders fail, we highly recommend trying ToddRivers\' sprite shaders.
They work with Spine-Unity. Make sure you choose the correct alpha blending option (Spine Skeleton shaders are Premultipled Alpha). Read more about Premultipled Alpha here.