The HR should explain the MBA sponsorship process in the first few weeks after you join: this information is very transparently available internally.
I can only speak for BCG and Europe, so I don't know if this helps: in our office you would have to wait 2 years and be a top performer to qualify for an MBA sponsorship. At the McK Germany office, on the other hand, virtually everyone could go. So that depends on the office, that's why you should inquire with the local HR.
There are a couple of MBA-related aspects that run across MBB firms and across the various offices, though. They are:
- you can only select from a list of up to 5-10 schools;
- you have to go back to your same office for two years after the program end OR repay the debt
- you don't have any salary advantages before your peers that didn't get an MBA
I hope this helps at least somewhat.