The link field uses the build in WP link insertion, the same thing that is used in the content area of the page. These links will break if the slug is changed the same way that links inserted into content will break if the slug is changed.
My first suggestion would be to not changes slugs unless there is a very good reason to do so. Bad for SEO.
My second suggestion would be to use a redirect plugin to add 301s when you need to change post slugs.
My third suggestion would be to use a post object field instead of a link field.
