That question is a little open ended.
Which CFI are you talking about?
To my mind you are talking about the CFI of the instructors normal Employer.
He cannot just go and moonlight for another company that doesn't have it's own AOC, hiding behind his own employers AOC. His CFI CAN give him permission to fly for another operator (actually cannot stop him), but the CFI cannot give him permission to hide behind that AOC.
The example i mentioned in my previous post, was a highly detailed process, that even saw the company offering the AOC services have to update its own Operations manual to include the types being operated by the new start up.
All instructors had to go through the AOC holders employee induction including SOP checks, check flights etc.
Once setup, it operates the same way as Royal vic was able to run Dual locations (YMMB & YCEM) under the one AOC.
Despite being quite willing to bag CASA, i have to commend the way they allow this to happen. It was almost a 6 month process for the start up company to get their own AOC, and it is purely a paperwork exercise. Without the ability for an AOC holder to umbrella another company in this manner, would probably see more GA training companies go bust.
I'm pretty sure the existing students at the said company were grateful for what transpired, as it ensured continuity of their training.
The downtime for training whilst the process to operate under the other companies AOC was less than 3 days.....and it pissed rain all week lol.