We introduce $p$-uniformity to characterize the scaling of density fluctuations in spatial random systems in $\mathbb{R}^d$, ranging from hyperfluctuation to stealthy hyperuniformity. Our central theorem establishes sufficient conditions to preserve $p$-uniformity under transport. The first condition, a finite $(d+p)$-th moment of the transport distance, allows for a Taylor expansion of the transport. The second condition controls the corresponding terms. We thus solve a previously stated open problem; indeed we extend it, since our result applies to a general $p$-uniform source in any dimension, and the source and transport may be dependent. As an application, we construct new classes of point processes that are isotropic and $p$-uniform with arbitrarily high $p$, and that can be simulated in linear time. We conclude with an outlook on a converse statement.