Normally, the shulker stays stationary. If you place a ship or harvester near the shulker, it will move into the ship. With the shulker in a boat or harvester, you can move it at the end.

To successfully teleport the shulker back to the overworld, you need to use a piston to push it through the portal. The entrance back to the upper world is located in the center of the island where the Dragon of ender was born. Here you can find a horizontal portal submerged in bedrock.

Carrying a shulker through a portal is a complex task, so check out this video on how to transport shulkers, as well as making a shulker shell farm.