Shulkers won't travel to other dimensions.
I expected to be able to force a shulker into the end portal or a nether portal once in the overworld for it to travel into the other dimensions. I tried pushing the mob with a piston, having the mob in place with a boat and then breaking the boat, and having the mob in place and then activating whichever portal.
Regardless of the method, or the dimension, the shulker would enter the portal and then disappear. It would not appear at spawn or 0,0 in the overworld or near the linked nether portal.
To reproduce this you only have to attempt to move a shulker through dimensions.
MC-168900 Shulkers teleport with original dimension coordinates after passing through portals