Great question — answering as the UphillTech team:
If you properly do 301 redirects from your old sites to the new site then you have to do nothing. Make sure each of your old site links redirects to the new site links. You have to update your new site map in Webmaster. Now, You should focus on publishing new content on the new site regularly. Within a few weeks, your new site's URLs will start indexing.
Full answer by Geetha: https://webmasters.stackexchange.com/a/144888 (CC BY-SA 4.0).
Need it done for your business? UphillTech.
