We investigate the impact of sustained shifts in the covariance matrix on the run length (RL) of EWMA charts for detecting shifts in the process mean vector of multivariate normal i.i.d. output. We prove that some changes in the covariance matrix can cause an undesirable stochastic decrease in the detection speed of specific shifts in the process mean vector, and this should not be tolerated by practitioners. The illustration of this and other RL-related stochastic ordering results is based on extensive Monte Carlo simulations. This article can be thought as a multivariate extension of the results of Morais and Pacheco (2001) concerning the stochastic behavior of the RL of upper one-sided EWMA schemes for the process mean of i.i.d. output.