If i am not mistaken search parameters (searchtext and searchmode) are passed by query string . Th e search results web part is looking for them. If you have 3 of thenm one page - they all going to be executed.
You have a user interface issue. 3 search boxes on one pages is a bit confusing. You need have one search box and drop where a user can pick search group (i.e. search index(s)). Essentially you need to form a query string like this ?searchtext=test&searchmode=anyword&searchindexname=CorporateSite.Default
And inside search result web part you put macro in search index text box: {%searchindexname%}.
So basically the only thing you need do is to add a drop down that will add/update selected index in the query string