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

My client recently launched a completely redesigned site. The new site and the old one are running in parallel at this time to help users make the transition.

I have a question about how Google chooses a default search URL for a website. At this time, if I write my clients' domain name in the Chrome address bar, followed by a space and then a keyword, Google will automatically search the site, using what it has obviously determined to be the URL of the site. Preferred search

This URL that you use is part of the functionality of the previous site, which will eventually be disabled.

What I would like to know, first, how did Google choose this URL and, secondly, what is the preferred way to suggest to Google that I should now use a different URL?

Obviously, I will redirect 301, for the old search URL when I turn it off, as I will with all the old pages of the site, but I wonder if there is a friendlier way with Google to handle this now, while the old pages of the site they are still active.

To cause minimal disruption to users, I would like to start directing as much traffic as possible to the new pages of the site now, before making the hard clipping.

Things I've done so far …

  1. The Google sitemap generator code was modified to include only the URLs.
    for the new pages of the site.
  2. Rel = "nofollow" was added to all links on the new site, back to the previous one.
  3. Added a rel = "canonical" link in the search of the previous site that points to the new one.

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