What flash diffuser is practical for travel with a Yongnuo YN560 III?

We have a couple of existing questions that might help. Take a look at When and how to use a push-on flash diffuser? regarding the plastic caps. They aren't really meant to be diffusers by themselves, since they are so small. Instead, they provide a bare-bulb effect, and if you are in a room with a low white ceiling and walls, the diffusion comes from light bouncing everywhere. They're no good outdoors or in big rooms (even though you'll sometimes see pro photographers using them this way; ah well). This is probably not very useful for travel, unless you know you'll mostly be traveling to small white rooms.

Next, see Are small on-flash softboxes useful, or a gimmick? The consensus is that they're not great, but better than nothing. So, you might try something there. I ended up getting the Honl Traveller8 Softbox, but I generally only use it for accent lighting as part of a multi-light setup; I've actually never tried it alone. If I were looking for something to use alone, I'd choose one with more area.

A little pop-up box like that is the best you're going to get on-camera, I'm afraid. For off-camera, which of course really opens up your possibilities for nice lighting, you're looking at carrying not just the modifier but a light stand or something as well. If you're gonna go for that, umbrellas are probably more portable than even the most portable softboxes.

Alternately, you could try something I just saw but can't recommend from personal experience: Westcott's Luna Grip, which is meant for hand-holding a remote flash in front of a collapsible diffusion disk of any size between 22" and 40". This would pack up fairly small, and you could use one of those 5-in-1 disks which includes a reflector, so you'd have other lighting possibilities, and you could press one of your parents or a friend or innocent bystander into lighting assistant duty, if need be.