When moving website from old domain to new domain it can be good to setup automatic redirect of old domain visitors & search engine robots to redirect them to new domain.
Following i found so called 301 redirect rule which should be added into .htaccess file which should be located in root folder of the website that should be redirected (usually in public_html or www folder)

.htaccess rule:
Options +FollowSymLinks
RewriteEngine on
RewriteRule (.*) http://newdomain.com/$1 [L,R=301]
in case i want to set old domain i can use this:

<IfModule mod_rewrite.c>  RewriteEngine On
  RewriteCond %{HTTP_HOST} ^olddomain.com$ [OR]
  RewriteCond %{HTTP_HOST} ^www.olddomain.com$
  RewriteRule (.*)$ http://www.newdomain.com/_sub/newdomain.com/$1 [R=301,L]
old.com/page.php -> new.com/page.php

Visitor should be redirected to proper page on new domain and robots do not see any duplicity. 301 is permanent redirect, so it should be probably telling search engine that they should update their index and stop considering old domain.

Similar: redirect old domain subpages to one page on new domain