  Motivated by mathematical tissue growth modelling, we consider the problem of
approximating the dynamics of multicolor P\'olya urn processes that start with
large numbers of balls of different colors and run for a long time. Using
strong approximation theorems for empirical and quantile processes, we
establish Gaussian process approximations for the P\'olya urn processes. The
approximating processes are sums of a multivariate Brownian motion process and
an independent linear drift with a random Gaussian coefficient. Which of the
two terms dominates depends on the ratio of the number of time steps $n$ to the
initial number of balls $N$ in the urn. We also establish an upper bound of the
form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex
Borel sets of the step $n$ urn composition distribution from the approximating
normal law.
