Leave the carts on the shelf in their boxes, which stack neatly, and use an R4? Or you could choose ONE system and take that with you, plus a few games. That should keep you going for a few hours... days, even, if you're on a trip!
Binders work. However, I keep mine in a GBA carrying case (there are plenty out there that do the same thing). Foregoing end labels, I just stack games in the order I got the games. The down side is removing and placing games back in their original spot is not as simple as opening up a flat case like those mentioned in previous posts.