There is nothing to stop you from migrating installed Linux to another PC as long as you understand it may not work in the new environment.
Here is a thread
I just wrote showing a cloned hard disk with 63 partition forcibly fed to another PC. 70% of the Linux boot successfully without alteration (apart from adjustment to the boot loaders).
You can certainly put the Linux onto a USB hard disk or even a USB thumbdrive.
If I were you I would copy the Linux onto a USB hard disk and convert its boot loader to boot from the USB hard disk. You can then carry the USB hard disk to any new computer and try to ask its Bios to boot from the the USB disk and have the Linux adjust to the new envirnment on a boot up. You can fill the USB hard disk up with any number of operating systems as demontrated here.
Ultimately I recommend load a few Live CD iso into a DVD or a USB thumbdrive so that there is several Linux you can use with any computer. A Live CD Linux is more compact and has been written to run on every PC. You need to keep you data in a separate partition which has to be mounted whenever you need them. This thread
may be useful as it describes how to put several iso into a re-writeable DVD and then on the thumbdrive.