Astrometric positions of radio-emitting active galactic nuclei (AGNs) can be determined with sub-milliarcsec accuracy using very long baseline interferometry (VLBI). The usually small apparent proper motion of distant extragalactic targets allow us to realize the fundamental celestial reference frame with VLBI observations. However, long-term astrometric monitoring may reveal extreme changes in some AGN positions. Using new VLBI observations in 2018–2021, we show here that four extragalactic radio sources (3C 48, CTA 21, 1144+352, 1328+254) have a dramatic shift in their positions by 20–130 mas over two decades. For all four sources, the apparent positional shift is caused by their radio structure change.