  We consider Gaussian approximation in a variant of the classical
Johnson--Mehl birth-growth model with random growth speed. Seeds appear
randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in
all directions with a random speed. The location, birth time and growth speed
of the seeds are given by a Poisson process. Under suitable conditions on the
random growth speed, the time distribution and a weight function
$h:\mathbb{R}^d \times [0,\infty) \to [0,\infty)$, we prove a Gaussian
convergence of the sum of the weights at the exposed points, which are those
seeds in the model that are not covered at the time of their birth. Such models
have previously been considered, albeit with fixed growth speed. Moreover,
using recent results on stabilization regions, we provide non-asymptotic bounds
on the distance between the normalized sum of weights and a standard Gaussian
random variable in the Wasserstein and Kolmogorov metrics.
