Most professors do some travel for their research. You might visit a department at another University, where you give a seminar talk and meet faculty there with related interests or with skills you don't have to solve part of a problem you are interested in. It is possible to answer really difficult research questions if you have the right collaborative team with overlapping skills. Such trips (as well as conferences) are a way to build up contacts to put such a team together. There is a certain component of these seminar visits of "showing off" your own work. If the people there like the work you present in your seminar, they will be more likely to work with you and your graduate students (and possibly hire them later), referee your research articles favourably and support your career by recommending you for research grants and promotion. This is an accepted form of self-promotion and we are, in general, happy to hear about new work in our field from visitors.
There is a spectrum of prestige for such travel, although even very good researchers do some travel at the low end. That is, you let someone know you want to visit their department and ask them to invite you to give a talk. You organize and pay for your own travel and accommodation (from a grant or your own pocket) and e-mail people there to set up meetings while you are there. Volunteering to be a speaker is often very welcome to seminar organizers as they try to fill their schedules with interesting talks. At the more "distinguished" end you are invited to come and they make the arrangements and pay. Even higher up (rarely, maybe never, in Mathematics) they fly you first class and give you a fat honorarium for coming to speak.
The highest end experience for me has been the Caltech Driver service. Caltech is a small, private school with a large endowment and a very high research profile. They have made a conscious decision to maintain a fleet of university vehicles and staff of professional drivers. I had this service on my recent trip, when a driver picked me up from UCLA and after my visit to Caltech I was taken to the airport. On a previous visit, I had the unique (for me) experience of arriving at LAX and being met by a Caltech Driver with a "Wetton" sign. It is a real luxury to have this service in the LA area where getting around by rental car or transit is quite stressful, especially if you don't know the city.
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