This is done both on the code side and in Spine editor.
In the editor, make animations that key body bones for running and jumping and whatever.
Also make an animation for swinging the sword, and the relevant bones that need to be moved. Key only the bones you need in those animations.
Then in code, you need to play the animations separately on top of each other.
The first parameter of the SetAnimation method determines a Track number.
If you play the running and jumping on track 0;
then the sword swinging on track 1.
You'll get the effect you need.
Any bones keyed by the animations with a higher track number will override bones keyed by animations playing in a lower track number.
// For Running
skeletonAnimation.state.SetAnimation(0, runAnimation, true);
// For Sword swinging
skeletonAnimation.state.SetAnimation(1, swordSwingAnimation, false);
Mix doesn't need to be touched at all for this.