Default location is ~/.adobe/Flash_Player/AssetCache but I would like to move it to ~/.cache
Would creating a symlink suffice? I.e.
Note this still stores the data at the original location and the symlink is at ~/.cache. You can also do the reverse if you first move the cache (make sure flash is closed when you do this):
Also as Eliah Kagan pointed out as ~/.cache already exists, it's better to place the adobe cache inside this location if desired.
I'm not sure whether adobe will let you change this setting - I've never come across this option anywhere.