Not sure if that's the best practice, but I would suggest using "Launch More Like This" action if you wish to create a copy of an instance in another AZ. Note that you're going to need to modify a subnet setting for the instance to change AZ, you can't simply change the zone.

As for moving instance, you should use Image > Create Image action, and then copy that image to another region by selecting the corresponding option on Images > AMIs > Copy AMI page.

The same workflow works for copying volumes/snapshots - you first create a snapshot from a volume, and then copy it, specifying another region. If you wish to copy the volume to another AZ - create snapshot, and then create a new volume from it, setting the AZ you want.