Last I've personally seen on it was from the "Letter from the Producer LIVE Part XXVIII":

Quote Originally Posted by Cherzy View Post
Just as I mentioned on a previous occasion, we'll be adding land in Patch 3.3, and we're aiming to have apartment-style housing in Patch 3.4. The moving feature will come after that.
We don't know how much "after" though so it sounds like it's at least 5 months away, if not more. We also don't know what it'll do it and if it's just going to be automating relinquish/purchase to ensure people don't end up with no house at all.

It'd be impossible to move furniture between different size houses/gardens so likely that'd all get sent to the Housing NPC (losing any breakable stuff because they won't want to give you that back, even binded). Whether it'll move walls/floors and airship progress is also unknown too.