The search is not being transferred. On the second screenshot you have 8 records in the found set on the web and 227 records in your filemaker form.
You assume that the search in the web form moves to a different record in the open FileMaker file, this is not true.
Something else is going on in FileMaker. Did you mention that there is another user using it localy?
Generally, if a FIleMaker file is shared, different users can run different searches, scripts and view/modify different records. The other users search results or records displayed will not be affected, unless you specifically script for this.
The file, which is broadcasting, should not be used directly, this is a bad practice.