A recalculation of the task should be automatically triggered, same functionality as a In-Project predecessors, when its cross project predecessor changes. This change should be reflected in the task update stream. Currently on the manually recalc the change is not recorded.
Tasks with a cross-project predecessor only update when you either manually recalculate the timeline, or wait for it to recalculate itself overnight.
(Documentation about it is here on paragraph 5, and automatic settings here)