With Yoothemes, the WARP framework compute the path incorrectly.
As the WARP library is included in each Yoothemes templates, module, components, extensions then we had to create a specific product (Patches for Yoothemes).
www.jms2win.com/en/joomla-multisite?page...46&category_id=1
It is not clear if this additional plugin and if the specific patches for widgetkit are installed (We think have understood that it is installed).
When a path is incorrectly computed by WARP, you have a full path of the master file location that might be present in urls (css, javascript, ...)
An alternative solution to fix the path is to use the "Search/Replace" plugin to replace the wrong path by the good path.
As you have noticed, each website should use its own cache directory.
The change that you did is wrong as the cache value could be wrong depending on the website.
Can you give us the version number of Widgetkit that you are using.