Ultimately the glue that binds movements together is proprioception. I like to think of it as lending quality to the movements we are using in training. Proprioception is awareness of joint position derived from feedback in the sense receptors in the joints, ligaments, tendons, and muscles. It is a highly trainable quality that we tend to take for granted. It is almost too simple.
Perhaps to appreciate proprioception we should look at the extreme case of a stroke victim that is able to return to normal movement patterns and full function. Why can’t an athlete who has all their capacities enhance the quality of their movement by focusing on the same things that the stroke victim has to focus on to get back to function? The key to that is proprioception. I think of athlete's as high level stroke patients. Look at the classical work of Knott & Voss in PNF for guidance, remember that their original work was with polio victims who were paralyzed so much of what they did was is in prone and supine postures. Take those principles and adapt them to standing and moving postures and you will have excellent transfer to performance. The more movement pathways we can open and connect the more efficient the movements will be and the better transfer to skill.
We can constantly change proprioceptive demand in a sensible manner throughout the training program in order to enhance the quality of movement. The actionable word here is sensible – no need to stand on a physioball or use other contrived positions, just use reciprocal and unilateral movements with resistance and assistance, close one eye, or turn your head away from the arm or leg that is moving. Simple modifications will heighten awareness and significantly change proprioceptive demand. Once again tap into the body's wisdom and learn to use it to improve quality of movement.