"When we switch a domain to mobile-first indexing, it will see an increase in Googlebot's crawling, while we update our index to your site's mobile version," Google Developer Advocate John Mueller wrote in a blog post. "Depending on the domain, this change can take some time. Afterwards, we'll still occasionally crawl with the traditional desktop Googlebot, but most crawling for Search will be done with our mobile smartphone user-agent."
He explains that the same "user-agent name" will match the Chromium version used for rendering. There are several ways to check whether a site is mobile-first indexed. One way is through Google's Search Console. Look for the status in the settings page, as well as in the URL Inspection Tool.
The announcement by Google detailing its work on mobile-first indexing, using a smartphone Googlebot to crawl the web, came in November 2016. The company spent a month experimenting on a small scale before implementing the change.
At the time of the announcement, Google Project Manager Doantam Phan published a list of recommendations to ease into the transition.
Some of those recommendations included serving structured markup for both the desktop and mobile version, and making changes to sites where the configuration shows the primary content and markup is different across mobile and desktop.