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The first two responses are typical of what I've seen on this cult--er, community. Thanks to Keri for answering the question so I can cancel and get back to ahrefs.

For wordpress.com you have to purchase the "Site Redirect" service.  This does a 301 redirect for you entire url.  You can go to Store > Domains in your dashboard to confirm it's working, change the new domain's url, etc.  Be sure to use the same permalink structure in your new domain!  Good luck.