This was reported for the "Add Node" search functionality, but is relevant in other searches as well.
So e.g. when searching for "Separate XYZ", typing "sep", then " " (with the intention to type "X" next) would clear the search field.
Now use the same method (matching against all search words) as in F3 searching ('menu_search_update_fn') in other searches as well [ID search template, objects as properties, finding nodes,...]
This should give a much nicer search experience in general.
Note: this doenst touch other searches in the Dopesheet, Outliner, Filebrowser or User Preferences that have other search implementations.