How does Google choose the default site search URL for the website?

My client recently launched a completely redesigned site. The new and old site work in parallel to help users make the transition.

I have a question about how Google chooses a default search URL for a website. At the present time, if I type my client's domain name in the Chrome address bar, followed by a space, then a keyword, Google will automatically perform a search the site, using the preferred search URL.

This used URL is part of the old feature of the site, which will eventually be disabled.

What I would like to know is, first, how did Google choose this URL, and second, what is the best way for me to suggest that Google now use a different URL?

I will obviously make 301 redirects to the old search URL when I turn off, as for all old site pages, but I wonder if there is a more friendly way for Google to handle this now, while the old site pages are still active.

In order to disrupt the users as little as possible, I would now like to direct as much traffic as possible to the new pages of the site, before making a sudden switch.

What I've done so far …

  1. Modifying the Google sitemap generator code to include only URLs.
    for the new pages of the site.
  2. Add rel = "nofollow" to all links from the new site, back to the old one.
  3. Adding a rel = "canonical" link in the old site search pointing to the new one.

These changes were made a few months ago, but Google continues to favor research on the old site.