When using a standard chain attack, you get to pick between your members for whose action happens. If you complete that action, you get to pick another (different) party member. Now, the tricky part that I don't understand is why I can sometimes pick 3 party members, and sometimes pick 4. Is there some reason behind why the game lets the chain go farther sometimes as opposed to others?
The chain attack meter can been seen in the lower right corner, and it depletes as chain orders complete. Whenever you run out of available characters or the gauge is entry, the chain attack ends.
For the standard 6 characters, the chain order will consume 1/3 of the gauge. However, Hero Orders will not consume any, merely using up any characters used during it.
Also noteworthy, an Ouroboros Order can be unlocked as a chain attack ends. Meaning, if you only have 2/3 of the gauge remaining, you can use it for 2 characters making up an Ouroboros pair. Then, even though your gauge will be empty, you'll still be able to do the Ouroboros Order.