The infinite-parent spatial Lambda-Fleming-Viot (SLFV) process is a model of random growth, in which a set evolves by the addition of balls according to points of an underlying Poisson point process, and which was recently introduced to study genetic diversity in spatially expanding populations. In this article, we give asymptotics for the location and depth of the moving interface, and identify the exact asymptotic scale of the transverse fluctuations of geodesics. Our proofs are based on a new representation of the infinite-parent SLFV in terms of chains of reproduction events, and on the study of the properties of a typical geodesic. Moreover, we show that our representation coincides with the alternative definitions of the process considered in the literature, subject to a simple condition on the initial state. Our results represent a novel development in the study of stochastic growth models, and also have consequences for the study of genetic diversity in expanding populations.