I think you make your own luck for things like this. I was a technology architect which had the advantage of being quite broad in scope. I came to the company with a deeper technical background than some which helped, but I also built a specialism after I joined that was rare in the company. That meant if there was a need for that skill, I had a high chance of being asked to assist. And of course, when it came to change projects, choosing ones that held an interest helped. But for example, anyone who joined the project with the police force had an opportunity to join a shift, anyone who worked on the TV project could have lunch where I did, so it isn't all about the role.
I'll chime in since I also worked in consulting for a long time (still do). It depends a lot on what vertical you get pulled into, but I'd say it's 50% luck and 50% you putting in the effort to develop and prove you have skills (including social ones) to get staffed on interesting projects. Many consultants never see more than conference rooms and the home office, but the parent commenter's experience also happens quite a bit. It's worth noting that in my experience the more advanced you get in your career the more likely you are to work only in a specific industry.