There is nothing required on your end to ensure search functionality when going live, everything will be brought over to and remain functional on the live domain.
When taking your new site live you’ll just need to update its nameservers and this should occur at the time of the site live, the Technical Support Representative processing the request will provide additional instructions on making this update when it’s time to make the change.
Once you submit your site live it will be processed within 5 business days. Prior to beginning the site live process, a Technical Support Representative will be in touch to let you know it’s commencing.
Here’s the link to our site live form: https://www.realestatewebmasters.com/site-live/
If you’re planning on transitioning completely to the new domain rather then having both active at once I would recommend setting up a redirect from the old domain to the new once your new site has gone live.