1) Clear ThunderBird Cache when it has been relocated in the following fashion:
Tools > options > Advanced > General > Config Editor
In the context menu, choose New to open a submenu and click on String.
You will be asked to enter the preference name, put in browser.cache.disk.parent_directory and click Ok. Next you will be asked to enter the value, which is the path you wish to use to store the cache.
It will create a directory called Cache inside the path you provide in the option, where it will store the cache files.
2) And the same for Firefox when relocated as follows:
In the URL type: about:config
Create a new string preference: browser.cache.disk.parent_directory
and specify a path to your cache folder, e.g.: F:\Firefox Cache