  Many growth processes lead to intriguing stochastic patterns and complex
fractal structures which exhibit local scale invariance properties. Such
structures can often be described effectively by space-time trajectories of
interacting particles, and their large scale behaviour depends on the overall
growth geometry. We establish an exact relation between statistical properties
of structures in uniformly expanding and fixed geometries, which preserves the
local scale invariance and is independent of other properties such as the
dimensionality. This relation generalizes standard conformal transformations as
the natural symmetry of self-affine growth processes. We illustrate our main
result numerically for various structures of coalescing L\'evy flights and
fractional Brownian motions, including also branching and finite particle
sizes. One of the main benefits of this new approach is a full, explicit
description of the asymptotic statistics in expanding domains, which are often
non-trivial and random due to amplification of initial fluctuations.
