We consider an operator introduced by Pfaltzgraff and Suffridge which provides a way of extending a locally biholomorphic mapping f∈ H (Bn) to a locally biholomorphic mapping F∈ H (Bn+ 1). When n= 1, this operator reduces to the well known Roper-Suffridge extension operator. In the first part of this paper we prove that if f has parametric representation on Bn then so does F on Bn+ 1. In particular, if f∈ S∗(Bn) then F∈ S∗(Bn+ 1). We also prove that if f is convex on Bn, then the image of F contains the convex hull of the image of some egg domain contained in Bn. In the second part of the paper we investigate some problems related to extreme points and support points for biholomorphic mappings on Bn generated using the Roper-Suffridge extension operator. Given a parametric representation for an extreme point (respectively a support point) generated in this way, we consider whether the corresponding Loewner flow consists only of extreme points (respectively support points). This generalizes work of Pell.