We start with a general introduction to Fraïssé structures and their automorphism groups. Of particular interest for us will be universal homogeneous ultrametric spaces. Here, we view ultrametric spaces as two-sorted structures consisting of a set of points and a linearly ordered set of distances, and equip them with distance-carrying ("dc" for short) embeddings. We prove that the automorphism group Aut(U) of the ultrametric space U obtained as a Fraïssé structure admits a comeager conjugacy class, or in other words, it has a generic dc-automorphism. To show that, we examine the cofinal amalgamation property for partial automorphisms and characterize amalgamation bases. Time permitting, we will outline a broader categorical framework for establishing cofinal amalgamation and discuss the non-existence of a generic pair of dc-automorphisms. This talk is based on recent joint work with A. Bartoš, W. Kubiś, and M. Malicki.