I think the key thing here is to always use relative paths in your blend files when linking remote data. That means for linked libraries, textures and any other data external to the file.
Relative paths are guarantied to work if you maintain the folder structure relative to the blender file intact, even if the absolute paths are changed.
That means if you move your files to a different machine or even on your own current machine but maintain their positions relative to each other things should not break.
Whereas absolute paths are never guarantied to work in a different machine, unless you can recreate the exact same base path for everything down to the same drive letter.
The good news is I think Blender already does this by default, so in theory you should not have to do anything, unless your store stuff across different drives letters, crosslink from network data, or specifically set blender to use absolute paths.
You can double check this by opening one of the files in question, going to the Outliner and changing the menu in the header to Blender File this should list all local data stored inside the current Blender file, and bellow it towards the end of the list all loaded external data from other .blends. From there you can check which files are loaded and from where.
If a path looks something like
//..\Path\to\my\Library.blend then is relative and all is good. If it looks something like
D:\Path\to\my\Library.blend (has a drive letter in it) than it is likely to break in the new computer.
You can easily fix this by going File > External Data > Make all paths relative
Blender has relatively good asset management capabilities, in the new computer you can then check if anything is missing by going to File > External Data > Report missing files
Now make sure that when you move stuff you maintain the same internal folder structure. One good way to organize your projects would be:
- Blender Projects Folder
- Project 01
- Project 02
- Project 03
- Project ...
- Library folder - For reusable assets and library files
Independently of where in the file system you place it, if you maintain this internal folder structure intact, and all files reference each other using relative paths within this "garden" all should be fine.