Google recently published a set of articles on how to transfer, move, or migrate your site. The information in these articles will be useful and relevant to you if you are moving your site, whether your domain name has changed or not. We recommend giving them a read.
There are various things to consider when moving a site, so we suggest that you familiarize yourself with the tools and documentation outlined by Google in order to make sure that your site move will be done correctly from their perspective.
If you have any questions regarding Google’s instructions as far as your WP Engine environment is concerned, please feel free to reach out to our support team. For SEO-specific questions, please refer to WP Engine’s SEO Recommendations before opening a live chat with Support, available 24/7, as you may find the answers you are looking for there.
If you have Google-specific questions, we recommend searching (or submitting to) the Google Product Forums; where you will most likely find the information you are looking for.