Moving my Steam account from old PC to a new PC and Library Sharing
It is impossible to transfer the shared library to another device.
You can lose your family shared library by replacing a major piece of hardware in your PC (a cpu or gpu, etc), because it is tied to your unique hardware ID, which is generated as the sum of all of your (important) hardware. While it is probably possible to trick Steam into thinking your new PC has that library shared, it would have to be re-authorized again, defeating the whole purpose.