Found it - with a little help from before.

After performing the merge, I needed to update each field individually:

Dim objStory As Microsoft.Office.Interop.Word.Range
Dim objField As Microsoft.Office.Interop.Word.Field

For Each objStory In docDestination.StoryRanges
For Each objField In objStory.Fields
objField.Update()
Next

' The header/footer ranges can be linked in a way
' that is not revealed by the outer For Each
' so we have to do the following
While Not (objStory.NextStoryRange Is Nothing)
objStory = objStory.NextStoryRange
For Each objField In objStory.Fields
objField.Update()
Next
End While
Next objStory

objField = Nothing
objStory = Nothing