To monitor count-type observations that are correlated over time, the Poisson INAR(1) CUSUM chart has been proposed for cases where an integer-valued autoregressive process of order 1 with a Poisson marginal is appropriate. In a recent study, it is shown that the Poisson INAR(1) CUSUM control chart is especially well suited to detection problems that involve mean shifts in a process. Although special causes of variation that result in mean shifts are the most common out-of-control situations in practice, some special causes of variation may also result in shifts of the autocorrelation coefficient or the process variance. In this research, we define three types of residuals from the time-series model and develop corresponding approaches for CUSUM monitoring. In particular, three CUSUM control charts, one for each of these residuals, as well as a multivariate CUSUM monitoring of the residuals are investigated. A Phase II comparison of these residuals' monitoring approaches together with a benchmark CUSUM of the count type observations indicate that, for different types of process shifts, different residual monitoring approaches may perform better.