Hello,
Here are my thoughts from my experience at B-School seeing my colleagues who were sponsored from MBB:
Part A: How does sponsorship work?
1- You certainly need to be a top performer to be able to receive financial support from the firm with a return offer. Typically you would have to work in the firm for about 2 years minimum (e.g. join as an Associate, do your tenure, and get sponsored for an MBA to return back as a Consultant).
2- There will definitely be a contract where you MAY be allowed to work at another company during your summer internship but you HAVE TO come back to the sponsored firm for a set duration of time after the MBA (typically 2+ years) or else you would have to pay back the financial component and risk damaging relationships.
Part B: Switching offices?
1- Its usually difficult. The sponsoring firm and office cares about your personal and professional development and would ideally like to reap the benefits of that through your continued tenure at the firm after your MBA
2- However, in exceptional circumstances (e.g. your partner lives elsewhere), they may allow you to transfer, albeit not immediately. I do not think leveraging another MBB offer from another office would always help.
3- You would also want to maintain positive relationships at your home office and may not be worth to burn bridges.
All the best!
Rushabh