There is no need to plug the bore or the touch hole when you are either browning the barrel or rust bluing it - you are wasting your time doing this
. If you are getting the browning solution in the bore then you are using too much of it
. As far as rust bluing the barrel it is placed in hot water after the browning solution has been applied to convert the rust oxides into the blue/black oxide color. Again no browning solution is in the bore. It would be the same as when you are cleaning your barrel with water after it has been shot.