WordPress has three methods for allowing “excerpts.” True excerpts are created by writing in the Excerpt field of the post – this is called a manual excerpt. In theme code, this excerpt is shown by using the Excerpt tag specifically: <?php the_excerpt(); ?>. When this is used, if you don’t have an excerpt in the excerpt field, WordPress will generate a 55-word excerpt from your post body.
The More tag is intended to generate “teasers” and is not the same as an excerpt. This only works if the template is using the full content tag: <?php the_content(); ?>.
Kiosk is coded to first look for a manual excerpt and if there isn’t one, then show the whole post. The Continue Reading link only applies where excerpts are shown, since the full content would not have anything to continue to.
So ideally if you are looking to customize the cut off of your excerpts, text should be copy/pasted into the Excerpt field instead of using the More tag. If you want to continue to use the More tag, you must customize it your own link which will show up where the tag is inserted. The following post on WordPress Codex has more info: