Fun with Cardboard
I keep each of my "fragile" prototype jigs in its own plain cardboard box, similar to the ones you get from Sparkfun or Amazon. (At my job I have plenty of leftover/scrap cardboard shipping boxes for our evaluation kits. Use what you can get locally.) Box should be tall enough to avoid the circuit getting crushed.
You can also mount the breadboard with double-sticky foam tape, on a flat piece of cardboard, trimmed to fit inside the box (maybe 1/4 or 1/2 inch smaller). That will keep it from shaking around inside the box during travel, yet you can still lift the breadboard out and put it on the bench for testing.
I also have used this technique when wiring up two or more printed-circuit board assemblies. A few #4-40 standoffs/nuts/bolts secures each board on the cardboard backing, which provides mechanical strain relief to keep the wires from breaking off.
Antistatic Bag (ESD protection)
I'd recommend keeping the board itself inside an
antistatic bag, inside the box. Besides the ESD protection, this helps contain any bits that might get shed off the solderless breadboard if the box takes a tumble. You can buy these bags online, or just collect them as most electronic components get shipped inside these ESD protection bags. (Although most parts nowadays aren't as vulnerable to ESD damage as the old CD4000 CMOS parts were back in the 1970's, the bags are cheap enough and shippers don't want to be blamed if parts arrive dead. So you may already have some of these, if you've bought your breadboarding parts online.) ESD damage can sometimes render a part fully dead, or worse, it can just damage the part a little bit and make the part not work correctly. So it's best to get in the habit of reasonable precaution against possible ESD damage.
Assuming you have a cellphone with a camera, it's worth taking a few pictures of the assembled breadboard... digital pictures cost almost nothing, and provide a good reference in case you need to rebuild or repair the circuit (or need to ask advice online).
Taking good, useful reference pictures requires a little bit of practice; try to rest your elbows on the table to avoid shaking the camera, and make sure there's enough diffuse light available (the camera's flash tends to make bright spot reflections that wipe out detail).