https://marco.org/2008/06/12/thatguyben-if-i-change-my-tumblelogs-url
If I change my tumblelog’s URL, what updates automagically and what doesn’t?
If you change your username (the “thatguyben” part in thatguyben.tumblr.com), people’s existing links to you at thatguyben.tumblr.com will break.
If you leave that the same and just add a custom domain (e.g. www.thatguyben.com), people’s links will still point to thatguyben.tumblr.com, but Tumblr will automatically redirect all of them to the corresponding page on www.thatguyben.com.