Since Phabricator (2019-07-03) / https://secure.phabricator.com/T5474 we have Triggers:
Every workboard column now offers a button in the column header.
It allows to define what to do "When a card is dropped into a column which uses this trigger". "Add projects tags / Assign task to / Change priority to / Change status to / Remove project tags" might be worth to document on https://www.mediawiki.org/wiki/Phabricator/Project_management
Also see https://phabricator.wikimedia.org/T91716 for historical discussions.
Lists of triggers is at https://phabricator.wikimedia.org/project/trigger/query/all/
Looks like anyone can create them. Not sure what that means for performance.