URL Change Question

Hi,

If I change the URL of a subpage on my site, will the old subpage URL just go away naturally, and not be indexed by the search engines after time? Or do I need to do something in WP dashboard?

When I changed the URl on the page, the old one still shows up in the SERP’s, but it’s blank with no text.

The new one shows in the SERP’s as well, but I’d rather have the old URL not be indexed anymore.

Thanks!