We consider invariant transports of stationary random measures on $\mathbb{R}^d$ and establish natural mixing criteria that guarantee persistence of asymptotic variances. To check our mixing assumptions, which are based on two-point Palm probabilities, we combine factorial moment expansion with stopping set techniques, among others. We complement our results by providing formulas for the Bartlett spectral measure of the destinations. We pay special attention to the case of a vanishing asymptotic variance, known as hyperuniformity. By constructing suitable transports from a hyperuniform source we are able to rigorously establish hyperuniformity for many point processes and random measures. On the other hand, our method can also refute hyperuniformity. For instance, we show that finitely many steps of Lloyd's algorithm or of a random organization model preserve the asymptotic variance if we start from a Poisson process or a point process with exponentially fast decaying correlation. Finally, we define a hyperuniformerer that turns any ergodic point process with finite intensity into a hyperuniform process by randomizing each point within its cell of a fair partition.