Moving or deleting pages should trigger update on Wikidata if page is linked through a property. Several properties on Wikidata point to pages on Commons and possibly other projects, for example [[ https://www.wikidata.org/wiki/Property:P373 | P373 ]] or [[https://www.wikidata.org/wiki/Property:P18|P18]]. Moving or deleting such pages on Commons should trigger update or removal of the property on Wikidata, the same way as moving or deleting pages connected through sitelinks is handled.
For example [[https://www.wikidata.org/wiki/Q13335|Q13335]]'s [[ https://www.wikidata.org/wiki/Property:P373 | P373 ]] was [[https://commons.wikimedia.org/wiki/Category:Universit%C3%A9_Bordeaux_4 |Category:Universit%C3%A9_Bordeaux_4]], when the category was deleted (for being empty), it [[https://www.wikidata.org/w/index.php?title=Q13335&type=revision&diff=409191463&oldid=342709365|triggered deletion of a sitelink]] but not of the property [[ https://www.wikidata.org/wiki/Property:P373 | P373 ]], which was latter [[https://www.wikidata.org/w/index.php?title=Q13335&diff=next&oldid=409191463| removed by a bot]]. Similarly [[https://www.wikidata.org/wiki/Q1169159|Q1169159]]'s [[ https://www.wikidata.org/wiki/Property:P373 | P373 ]] was [[https://commons.wikimedia.org/wiki/Category:Warmbad_Sellin |Category:Warmbad_Sellin]], when the category page was moved to [[https://commons.wikimedia.org/wiki/Category:Warmbadstra%C3%9Fe_4_(Warmbad)|Category:Warmbadstraße 4 (Warmbad)]] it [[https://www.wikidata.org/w/index.php?title=Q1169159&type=revision&diff=376706768&oldid=373873652 | change of the sitelink]] but not the property which remained broken until today.