Posts go from draft to published to trash (bin) and then deleted. Any of these actions triggers a cache build/re-build for the post in question. I do not know, but my guess is that calls to admin-ajax.php are what does it. Some setups will cause changes that Search & Filter does not detect, and thus if you find that happening, you might need to add code to whichever plugin is doing that to trigger the cache to build, and this is covered in our documentation (in the actions).
A post is only fully removed in WP when it is deleted from the bin. In the S&F form settings, you specify what state the posts it searches are in (usually ‘published’).
As to why the cache does not build in the background, you may need to increase the cache build speed in the plugin settings, or it may be the PHP settings of the server that need adjusting (such as timeout settings).
.
