The fourth batch, I believe that is what we are up to now, of notices that sites have been moved over to the mobile-first index has gone out over the weekend. I received a number of notifications over the weekend, like the previous batches, none of the sites I saw moved over were actually mobile-first – but we explained why that is the case over here.
If I had to date these batches, they would be on these dates:
March: Batch one was a small subset back in March
April 30, 2018 – was the second batch, but was way more wide spread
May 17, 2018 – was the third batch of notices
May 27, 2018 was the fourth batch of notices
Joost de Valk from Yoast noticed a number of sites getting this fourth batch of notices as well:

There’s a new batch of mobile indexing enabled e-mails going out, have had a few in the last few minutes. /Cc @rustybrick
— Joost de Valk (@jdevalk) May 27, 2018
Here is the message from the fourth batch:

Mobile-first indexing enabled for [domain]
This means that you may see more traffic in your logs from Googlebot Smartphone. You may also see that snippets in Google Search results are now generated from the mobile version of your content.
Background: Mobile-first indexing means that Googlebot will now use the mobile version of your site for indexing and ranking, to better help our (primarily mobile) users find what they’re looking for. Google’s crawling, indexing, and ranking systems have historically used the desktop version of your site’s content, which can cause issues for mobile searchers when the desktop version differs from the mobile version. Our analysis indicates that the mobile and desktop versions of your site are comparable.

Forum discussion at Twitter.

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