Exact behaviour of CHANGE_TRACKING AUTO
concerning "The full-text index is updated in the background, however, so propagated changes might not be reflected immediately in the index." from what I can tell and experience it means the change to the full-text is updated as any other clustered index, using a background task and not on the immediate instant that the data was changed and the update was triggered.

Also reading a bit more on the documentation (create fulltext index):
"CHANGE_TRACKING [ = ] { MANUAL | AUTO | OFF [ , NO POPULATION ] }
Specifies whether changes (updates, deletes or inserts) made to table columns that are covered by the full-text index will be propagated by SQL Server to the full-text index. Data changes through WRITETEXT and UPDATETEXT are not reflected in the full-text index, and are not picked up with change tracking."

This is for the manual option meaning that for the AUTO the reverse can be applied, the change is done by the change tracking event and not immediate.

And here is why (Population based on change tracking):

"Optionally, you can use change tracking to maintain a full-text index after its initial full population. There is a small overhead associated with change tracking because SQL Server maintains a table in which it tracks changes to the base table since the last population. When you use change tracking, SQL Server maintains a record of the rows in the base table or indexed view that have been modified by updates, deletes, or inserts. Data changes made through WRITETEXT and UPDATETEXT are not reflected in the full-text index, and are not picked up with change tracking."