The OP is based in the UK, not the US.
I'm a mergers and acquisitions lawyer who has lived and worked around the world, as an employee solicitor and a partner of large international law firms. Here's my two cents on options for travelling as a solicitor in private practice:
1. If you are admitted in England and Wales (or New York) this gives you an increased opportunity to travel for work because international transactions (ie those with a cross-border element) are very often governed by English or New York law.
2. Don't pick a practice area that is jurisdiction-specific. Property law is a pretty good example because the law is usually specific to your "home" jurisdiction and is not necessarily regarded as "translating" well to other places. It's not impossible, but it's harder than in other areas. Your extensive knowledge and experience of English real estate law, for example, would be basically irrelevant if you want to work in, say, Norway.
3. Litigation can be limiting in the same way. Some litigation solicitors/attorneys do a lot of travel if they act for multi-national clients or happen to get a huge matter with an international element, but on the whole they are confined to base (or limited travel within their jurisdiction) IME.
4. Areas that could be worth checking out tend to be transactional in nature (ie they service clients doing big corporate deals, large infrastructure projects etc) - for example, banking, structured finance, project finance. Banks do business all around the world. Infrastructure project developers and pension funds can be the same. Mergers and acquisitions (although the market is not strong right now) can provide opportunities, again if you cultivate an international client base.
5. Get a training contract at a law firm that has offices around the world. You probably already know that there are numerous leading English firms that fit this description. These firms may provide opportunities to live overseas for short stints during your training contract and also longer international transfers down the track as well as day-to-day travel for clients. These positions are very sought after so make sure your marks are strong and you have lots of good experience on your CV.
6. Get some experience in a City firm - ie London not Manchester - during your law studies (you probably know about clerkship opportunities already) and talk to as many travelling partners as you can about what they do. This will significantly improve your chances of securing a training contract in the sorts of firms you might be interested in.
7. Whether or not you travel, the practice of law has a large, unavoidable amount of desk-based work. If you want to be in an area of practice that offers travel opportunities, you can expect to work demanding hours because that is the nature of the deals that tend to require travel. Like most work-related travel, it's not all glamour and excitement. There would definitely be times when you would feel stuck behind a desk 24/7! If you are concerned about being stuck in a desk job then I would definitely try to get some work experience ASAP to see if the law really is for you.
Good luck with your studies.