In the mathematics milieu, there'd be little motivation to do this, and several motivations to not do it. E.g., if one is in a top-ranked program, contact and connections with the faculty there, e.g., one's thesis advisor, are critical, not only to eventually generate letters of recommendation but, presumably, to be exposed to their ideas, their ways of thinking, and many intangibles.
A scenario in which I could imagine "travel abroad" would be professionally useful would be in which one found oneself in a "second tier" program, but somehow had the opportunity to "visit" a "first tier" program for a substantial bit of time. Then the point would be to pay attention to the ideas and viewpoints of the faculty there... obviously... and eventually elicit letters, maybe?
Otherwise, merely dislocating oneself from one's base is pointless.
Edit: quite apparently (in light of JeffE's comments and whoever's downvotes and such) there are varying viewpoints on the benefits-or-not of "study abroad". Presumably this is related to one's mental model for what happens in grad school, especially the role of the advisor, but also the "maturity/responsibility" of grad students. Also, while "travel is broadening", depending on one's model "broadening" may not be the goal of grad school. Sure, the extent to which "travel" makes provincial prejudices harder to maintain, one might hope that being at one of the best places avoids that already. If one's model views advisor/student as master/apprentice, disconnection seems undesirable. If the model views the advisor as merely an older colleague who's been successful, then very different actions seem reasonable. The latter sort of model-feature is arguably a corollary of the "grad student as independent thinker" principle. (My preferred tweaking of this is to "critical thinker".)
To my mind, the bottom line is that some of the most interesting projects/issues have enormous and informative backstories very badly documented in the formal literature, so that even a very good grad student has tremendous difficulty assimilating things. Optimistically/ideally, one can get different perspectives on these issues by talking to different "top experts", hence motivating "travel". However, a key bottleneck is that it may take a few years of full-time attention to catch on to a mature expert viewpoint. If there were a unique, objective such, then one could get variations on it from various experts. However, it is not at all clear to me that there is such a unique, objective unifying "story".
In fact, the variations on what might have been "the standard story" appear to often be fairly critical, and the variations and nuances thereof take a long time to get a grip on. Thus my raising the "apprenticeship" model.
As hinted at, if grad school does take "several years", maybe there is more room to fit in residence elsewhere, but some of the elite programs like people to finish in three or at-most-four years, and then it's harder to see how this would work out.