In you descriptions it sounds like users are taken to a google search result page for Search term site:ourbrand.com
If that is the case, no this in not normal behaviour if you have implemented the sitelinks search box schema, as you have done. This is the default behaviour if you haven't implemented the schema.
If the schema is working correctly, the user should be redirected from Google search pages to the search result page on your site.
Looking at your mark up, that should be
Looking at your schema mark up example, it all looks correct. Perhaps the search result URL from your site is incorrect? Also if you haven't already, double check the mark up in Google validator: