While some column types allow this, there appears to be no way to use date.
When a date column is mirrored, it currently can’t be used in an automation to trigger an action. The automation simply doesn’t see the column although it, obviously, contains a date.
As an example, I’m working in a team where we have a launch date that is set on one board as a source of truth and then gets mirrored across other boards. There appears to be no easy way to just pick up that column in an automation which would be very useful and also prevent any manual changes and duplications.