compared to McKinsey, BCG is operating more in "regional cells". This means, they are less globally integrated as a firm, which has a couple of implications of global mobility of consultants.
Generally speaking, there are several types of transfers - and it becomes trickier, the more "permanent" you want your move to be.
Temporary office transfers:
Lasting 12-18 months, you're essentially an expat, and you're expected to transfer back to your home office afterwards. One tricky twist (at least as of a few years back) is that the whole exchange program is supposed to net out, ie some other consultant needs to be found to take your place. As you can imagine, this is easy if you're based in a very popular location (like London or NYC), trickier for some other offices.
Some offices, especially the ones that have been setup recently and are growing fast with a lack of local talent pool have programs to facilitate temporary or permanent office transfers. It should be pretty easy for you to get to know which ones are currently "open" right now. Of course, by definition you're talking emerging countries and/or relatively exotic geographies - not NYC or London.
If you want to move permanently:
If you want to move somewhere else on a permanent basis, that's a bit trickier with BCG. Given that they operate less in a global "one firm" way, it is not in your home office's best interest to let you go, because they were the one to invest on finding you, firing you, and training you. So, they basically won't have an inclination to help much - instead they might try to have you wait. So, if this vital for you, being crystal clear about what you want will be critical to make it happen. You will literally have to go on peoples' nerves (CDA, HR, your current managers). Even more important: reach out to your target office to sell yourself! Concretely, to make the transfer happen, you have to make the target office pull you!
All of that being said, ther is one overarching truth to it: if you are a high performer, everything will be MUCH easier. If you struggle, you might not be a ble to pull it off.