Braiding a link with a fixed closed braid

We show that every oriented link diagram with a closed braid diagram as a sublink diagram can be deformed into a closed braid diagram by a deformation keeping the sublink diagram and, under a mild condition, the number of Seifert circles fixed. As an application, we give an upper bound for the braid index of the link obtained by reversing the orientation of its sublink by using only the information of an original link.