I feel like it has to do with bending someone’s will to your own. The advanced forms we’ve already seen appear to be extensions of the color’s basic functions - CoA allows you to harden your body, advanced CoA lets you use that hardening to affect your opponents’ bodies; CoO lets you observe things at a distance, advanced CoO lets you observe things a little into the future. Therefore, if CoC allows you to overcome weaker willed enemies, stands to reason that advanced CoC allows you to take control of their spirit entirely. How that manifests is entirely up to the user - whereas Kaido and Linlin might use it to force people to swear fealty to them, Luffy may unconsciously rally those around him to his cause. We’ve seen this play out many times already, with Mihawk noting it as the most dangerous ability on the seas during the war at Marineford, so I believe that’s a result of advanced CoC.