Further information has been offered by TelerikSupport:
When the page changes the url is also changed. Edit the search results widget and go to Advanced-UrlKeyPrefix and enter the prefix "search", for the other widgets on the page also go to edit them and select Advanced ->UrlKeyPrefix and set the same string "search", now all widgets will share the same url.
Ok I had a try at that but unfortunately my SF version must be different to yours so I am seeing different properties. I am using SF 6.1.4600 here on all versions of my site. When editing the page template which contains my custom widgets, the properties
dialog does not contain the properties you mention as there are only three (AsyncManager, Profile and ViewEndingCollection). I presume these are the default for a custom MVC widget.
When I edit the SearchResult page which has the standard SearchResult widget on it, and open the properties of the widget, there is no property named UrlKeyPrefix. There are ones called PageKey and UrlEvaluationMode but I dont know if these are what you meant.
TelerikSupport came back with:
I am sorry for missing this, the UrlKeyPrefix is not available as property for the MVC widgets to be added they need to add an attribute to each ActionResult as explained in this blog post, however the search results widget will not match the added attribute because Search results when rendering the links to search results pages doesn`t have an out of the box way to set UrlKeyPrefix.
Custom MVC widgets disappear when the paging have been changed because they are dependant on the page route and when the route changes to display a page the widget action is not executed.
I will research and find a way to render the MVC widgets when the search have paging by either making a modification to search results widget or adding specific case of handling the route in MVC widgets. I am sorry for the delay.