TL;DR: Highlight your complex structure, hold "Use" (E key by default), move mouse. Whole structure moves.
There is a slightly obscure mode for modifying structures in Fallout 4. On the PC, it involves holding down the "E" key for "Use". (Someone else needs to test on consoles.)
If you have an object "picked up", holding down the "E" key will allow the mouse to move just the furniture, the mouse scroll will control its elevation. This will obviously only move one piece at a time, but it is still very useful.
The second mode is without any objects picked up, highlight an object and hold down the "E" key. After a short while, a set of objects will be "picked up". Fallout 4 will then try to move all the "attached" objects together in one giant set. This may sometimes pick up nearby turrets and objects, so you may have to store them.
(I'll have to make a video later because this mode just blows my mind. I can now setup complex stairs in the open and then move the whole thing into position.)