Hi Mark,
The Breadcrumb isn't actually closing the drop-down. It just uses a native WPF ComboBox with a custom style. The ComboBoxStaysOpenOnEdit property simply sets the ComboBox.StaysOpenOnEdit property, which only keeps the drop-down open when the user clicks the TextBox control in the ComboBox (not when you select something from the drop-down list).
The ComboBox.NotifyComboBoxItemMouseUp method is called by the ComboBoxItems, and is what is closing the drop-down. You may be able to add a PreviewMouseLeftButtonUp handler to the Breadcrumb for the ComboBoxItem type, and then handle the event yourself (i.e. set the selection of the associated ComboBox).