It is near impossible to get a job in the the disputes practice of a white shoe firm after an LLM, even if it is from Harvard or Yale. But there are quite a few folks who have transitioned to a career in Big Law, although most of them practice transactional law. A fair few of them get reassessed after their first or second year (typically a few months after they have successfully completed getting barred considering first year associates are termed law clerks and not associates until they complete all licensing formalities). Having said that, its not impossible -- I have heard of at least one senior from law school who practices US litigation and international arbitration at a top NY shop. But then he is an exception in many ways -- guy is probably overqualified for the job considering he was up for partner at his previous firm and is a quite senior in the profession and generally well-known in the community.

If you are interested in international arbitration, then I recommend studying in Europe as it is generally cheaper than the US or UK, tends to have more scholarship/aid, and allows you to intern with the firms there, which are generally more open to foreign lawyers, especially if you are skilled in English legal writing.

Speaking as someone who came back to India after doing an LLM, I would say dont do it if you intend to practice international arbitration. There is barely any work to go around and I cant say people are professionally treated. There is crazy oversupply and standards have fallen long and hard. It is probably the only practice area that attracts qualified and experienced lawyers who are coerced into accepting minimum wages internships/traineeships for (increasingly many) years. And thats only the beginning, prospects only dip lower as one ascends the law firm ladder, where "native" lawyers who can also litigate having a distinct edge over others. Even compared to other practice areas, it is well accepted that in international arbitration the hours are brutal, drafting is unskilled and unnecessarily prolix, and in general you get -- forgive me if I'm getting my metaphors twisted -- much more "bang" for the buck than your peers.