I go to a lot of schools, homes, library locations, etc. with my VEX parts. I’ve been taking small tubs of parts but spent a ton of time rummaging around in them to find things.
I bought a small rolling suitcase to carry things around. I chose a hard sided case to protect things inside, but a soft sided case may work for you. The top has a zipper pocket, I can keep copies of the instructions and a sample notebook. You might want to pick a soft side version that will give you extra pockets on the outside.
Inside it holds 7 of the small parts drawers that you can pick up from Harbor Freight for about $3 each on sale. The larger box holds larger parts that won’t fit in the smaller tray (from the Container Store) I’m able to carry a few of each VEXIQ part, along with a demo bot. Shown is a simple 4 wheel bot, but the Clawbot / Stretch will fit if you can cut back to 6 trays.
Case has those multi rolling wheels so it moves along pretty easily down hallways. Bonus is feeling like George Cloony in “Up in the Air” as you transport all your IQ parts from place to place.
Pro buying tips: Take two of the parts boxes with you to the store to make sure they will fit. You may need to look at cases that expand (mine does to fit the boxes). Make sure the slide in/out handle works well and pulls out far enough to make it easy pull the case. Plan to spend time in the store since most places seal the cases to prevent loss by people getting a suitcase and wheeling it out full of stuff.
I’ve had mine since September and it’s one of the best purchases I’ve ever made.