Continuing from Note6, as branch-points are moved by varying parameters let an associated system of cuts be moved with them.
Now assume that there are two branch-points such that a suitable parameter variation causes them to merge to become one branch-point. Use a system of cuts in which they are adjacent. On merging the cut joining them ceases to exist. Take a circle centred on the joint limit and so small that only the two which are to merge enter it. For each other cut take a fixed contour avoiding the circle and crossing that cut only. If the non-merging branch-points move under the parameter variation the contour has also to be kept ctear of their tracks. This is a detail which is left to the reader. Then as explained near the beginning following the contour from initial point back to initial point gives a permutation on branches  which is assigned to that cut. So under merging of two branch-points the surviving cuts retain their associated permutations.
Note that if the original system of cuts was standard the derived system will also be standard since the number of branch-points and the number of cuts are both reduced by one.