Time series models are often constructed by combining nonstationary effects such as trends with stochastic processes that are believed to be stationary. Although stationarity of the underlying process is typically crucial to ensure desirable properties or even validity of statistical estimators, there are numerous time series models for which this stationarity is not yet proven. One of the most general methods for proving stationarity is via the use of drift conditions; however, this method assumes phi-irreducibility, which is violated by the important class of count-valued observation-driven models. We provide a formal justification for the use of drift conditions on count-valued observation-driven models, and demonstrate by proving for the first time stationarity and ergodicity of several models. These include the class of Generalized Autoregressive Moving Average models, which contains a number of important count-valued and nonlinear models as special cases.