On the USB3 drive, create a folder called Template Project. Inside that folder, create at least the following three folders: Media, PR Project Files, Exports.
For each project, duplicate the Template Project folder and rename Template Porject Copy to reflect the current project name.
When starting a new project file in Premiere Pro, set all of the Scratch Disk Settings to “Same as Project” and save the .prproj file to the corresponding PR Project Files folder created earlier.
Before importing any source footage, make sure it’s in the corresponding Media folder created earlier.
Everything will be on the USB3 drive and you’re good to move around as needed.
Or course, make sure you’re using the same version of Premiere Pro an each host workstation as to not accidentally update the Premiere Pro project file.
-Warren
Oh, almost forgot! If working cross-platform is required, make sure that the USB3 drive is formatted as ExFAT. If you go with NTFS, then the Mac side will need Paragon installed for cross-plaform. If you go with HFS+, then the Windows side will need MacOpener installed for cross-platform. Hopefully your lab already has Paragon and/or MacOpener as you never know how student drives will be formatted.