100% possible! In fact, I've just coached 2 candidates from big 4 into MBB! Interestingly enough, I mostly see moves from Deloitte specifically.
These moves also tend to happen at the manager level (rarely below).
When this does happen, remember that you're being hired for your expertise/specialization. So, make sure to apply as a specialist (i.e. specific industry/function). For reference, my 2 most recent candidates were being hired in the Energy practice and the Digital/IT practice, respectively.
The interview process is absolutely different. You may get cases, and, if you do, they will be specific to your industry/function. If you do not get cases, you will have a number of "brainstorming" questions and unstructured cases...they're testing not only your critical thinking and communication abilities but also your industry/function know-how.
Feel free to reach out for any additional clarity/questions!