Wikipedia says building were being moved since 19th century. We also had a large-scale building relocation during Stalin's Moscow Reconstruction (basically, destroying beautiful historical houses to expand ancient curvy streets into straight wide roads) - some buildings were moved without cutting water supply, electricity and even with residents inside [1]. This probably was intended to showcase the achievements of Soviet engineering but judging by the comments here (and by English Wikipedia article on the topic) nobody even remembers this.

Personally I think that there should be no new construction in historical areas and there is no need to move anything.

[1] https://www.mos.ru/en/news/item/13744073/