This paper proposes strict periodic stationarity and periodic ergodicity conditions for a finite mixture integer-valued GARCH model with $S$-periodic time-varying parameters that depend on the state of an independent and periodically distributed regime sequence. In this model, the past conditional mean values depend on the past of the regime variable in the same order, so the model is characterized by path-regime dependence. We also propose sufficient conditions for periodic stationarity when the conditional means are nonlinear of past observations. The results are applied to various discrete conditional distributions.