As long as the directory structures of the library / my sets folders are the same each side, there should not be a problem. I run Live on PC and Mac; the library and set on each are synced via the cloud( Dropbox) ; the directory structure and files are exactly the same on each, and so far I have had no problem.
can a mac owner verify if ableton use .asd's on OsX?
yes it does
This may in fact be the source of the problem: If you installed the library fresh from the disk, you may have the problem that Live never installed the little .asd files that have the clip information. Live generates them as you work with clips, and if you do a fresh install you won's have them
If you can, do the following: PHYSICALLY copy over the entire library from your PC to your new machine. Don't install it from the disk, copy the files over from the PC.
Then set live's file path in preferences to the copied library.
When you open a set for the first time, it may tell you samples are missing.
Direct the automatic search to your sample library and it will search for them there
It will take quite a while search and find all the files, but then you should be sorted.