Well, I don't think I'm that far from the solution because the following code can display the contents of my listbox!
(Then, I'm not sure I'm using the right vocabulary... but apparently they're ContentControlListEntry since I can display each item in my "list")
Code:
Public Sub _2_browse_through_all_the_listboxes_in_each_document(I, Fichier_Objet)
Dim objCc As ContentControl
Dim objLe As ContentControlListEntry
Debug.Print Fichier_Objet
For Each objCc In wordApp.ActiveDocument.ContentControls
For Each objLe In objCc.DropdownListEntries
Debug.Print objLe.Text
Debug.Print objLe.Index
Debug.Print objLe.Value
'Debug.Print objLe.Creator
Next
Next
End Sub