The propagation of the error covariance is an important, but not well-understood, aspect of the statistical estimation of dynamical systems, such as data assimilation. Motivated by atmospheric data assimilation, this presentation considers the problem for states governed by the continuity equation and related hyperbolic partial differential equation. Careful analysis of the continuum and discrete problem demonstrates that standard methods of discrete covariance propagation are inherently inaccurate. The underlying problem is caused by hyperbolicity: the diagonal of the kernel of the covariance operator is a characteristic surface for advective dynamics. Based on the insights gained from this analysis, an alternative method of covariance propagation is proposed that directly addresses the problems that arise in conventional propagation methods.