We get this question a lot but I don't think anyone's ever specifically asked about traveling and that's pretty cool! And I do have an answer! :) The incomparable Museum of Menstruation has an answer for the 19th century in the Western world, which is that women of means when traveling, used disposable pads instead of the more customary reusable ones you'd have at home, and burned them to dispose of them, probably in their hotel room's fireplace, but possibly in a little brazier you could travel with. His site is a little rough, but there is good primary source info there, and he cites his stuff pretty well. The first ad on this page makes explicit instruction that these special disposable pads are to be burned when traveling. (Burning trash is kinda out there now, but would have been more common in the past, especially before modern city sanitary services.) This is pretty expensive though, and there is also record that women took their normal cloths when traveling and just bagged them up to be washed at home by them (or their maid) in a waterproof bag of some sort. It's also hard to believe this sort of problem really existed before the 19th century, as women (especially women rich and dainty enough to worry about surreptitiously burning their dirty pads) traveling and staying in public accommodations would have been pretty unusual before the invention of the "hotel." If you were a lady traveling and staying with friends and family for an extended time you'd have presumably just managed your blood as normal. This solution is a pretty weird slice of history, caused by the small overlap between women starting to leisure-travel but before disposable pads (and later tampons) becoming the custom over reusable ones.

For earlier periods (heh), he is of the opinion that the majority of women free-bled into their clothing and used nothing in particular. Part of his argument is that, prior to modern nutrition, puberty onset, and our customary long delay of reproduction, a menstrual bleed would have been less common of an event for women, and no where close to the "monthlies" we enjoy now. Also consider the volume of the average period is only about 2 or 3 tablespoons, which is really not that much liquid for fabric to absorb, especially spread over 3 to 5 days. I'm not sure I totally buy it though, as in Western dress fashion women didn't wear panties as we know them then, with a closed crotch, and it would dribble down your leg and probably get all on your stockings, and that's a bit much for me to believe most women just went "oh well" about. It's a compelling argument though, and worth considering. But there was probably a variety of options that women used, depending on their taste, lifestyle, and how much blood they produced, but it just hasn't been well recorded. There's also a pretty big overlap (dare I use the phrase throughout history??) between how a society diapers babies (if they diaper, not all societies do) and how women deal with periods. Like this fine tech today is used in both diapers and pads, and is what makes possible the "ultra thin" pads, excluding the Always Infinity pads which are a different tech. Some societies in America would have used moss for both. Look to the babies, and you get a big clue of how women probably perioded. (Uh, aside from the internal options, obviously.)