Let me expand this question a little bit to "what to do when you're rejected" from one or all firms you apply to.
First of all, I hate to break it to you, but in many cases it is not "just" the case study part of the interview. The case is the easiest to give you feedback about, rather than saying it was no match because of lack in personal fit or some other point.
It may also be interesting to some of you that there is a thing called "reject re-apply". It can happen that all interviewers really liked the candidate and that she seemed really smart and capable, but there were some things about the case that just weren't good enough (e.g., bad business judgment), or the candidate seems a bit too young/naive but shows great potential otherwise. "Reject re-apply" basically means -- we won't hire you right now, but we think you have a lot of potential and if you build more experience over the course of ~1 year, happy to take your application again. It is rare and I am not sure if all firms do it, but some do. That in general is a positive sign, even if it may feel bad at the moment.
So, what if you are rejected without given the explicit outright opportunity to re-apply? First of all, I would try to look at more / other firms. The big firms often have smaller subsidiaries / side activities where it may make sense to apply to (e.g., BCG has BCG Digital Ventures). Smaller shops/firms often are another alternative.
Regarding international applications (your original question): As far as I know, most do not share applicant information (not even sure if they are legally allowed to do that). However, you should show some rationale why you are applying in that specific market or country. I know that if that connection is more difficult to make or even for practical purposes, they may ask you to apply (or do the interviews) in your home market. So, to make it credible, you should at least have some story why you now apply in the different market (e.g., your significant other moved there or something). The situation gets even more complex when you require a visa to relocate, in which case your home country's office may be involved some way or another anyways.
After a certain period, you can also just apply again by yourself in your home market. I would wait for roughly 1.5-2 years before doing so, and know several successful candidates who have done just taht! Important for you would be to show some kind of professional growth in the meantime, which explains why you should be looked at or judged differently now.
To sum it up - why not give it a try :) But be prepared in case they find out and call you out on it. Good luck in any case!!