A new integer-valued autoregressive process (INAR) with Generalised Lagrangian Katz (GLK) innovations is defined. This process family provides a flexible modeling framework for count data, allowing for under and over-dispersion, asymmetry, and excess of kurtosis. It also includes standard INAR models such as Generalized Poisson and Negative Binomial as special cases. It is shown that the GLK–INAR process is discrete semi-self-decomposable,​ infinite divisible, stable by aggregation and stationarity conditions are provided. Some extensions are discussed, such as the Markov-Switching and the zero-inflated​ GLK–INARs. A Bayesian inference framework and an efficient posterior approximation procedure are introduced. The proposed models are applied to 130 time series from Google Trend, which proxy the worldwide public concern about climate change. New evidence is found of heterogeneity across time, countries and keywords in the persistence, uncertainty, and long-run public awareness level.