(Abridged) The formation and evolution of star cluster populations are related to the galactic environment. Cluster formation is governed by processes acting on galactic scales, and star cluster disruption is driven by the tidal field. In this paper, we present a self-consistent model for the formation and evolution of star cluster populations, for which we combine an N-body/SPH galaxy evolution code with semi-analytic models for star cluster evolution. The model includes star formation, feedback, stellar evolution, and star cluster disruption by two-body relaxation and tidal shocks. We apply the model by simulating a suite of 9 isolated disc galaxies and 24 galaxy mergers. The evolutionary histories of individual clusters in these simulations are discussed to illustrate how the environment of clusters changes in time and space. The resulting variability of the disruption rate with time and space affects the properties of star cluster populations. The combined effect of clusters escaping their dense formation sites (`cluster migration') and the preferential disruption of clusters residing in dense environments (`natural selection') implies that the mean disruption rate of the population decreases with cluster age. This affects the slope of the cluster age distribution, which becomes a function of the star formation rate density. The evolutionary histories of clusters in a galaxy merger vary widely and determine which clusters survive the merger. This impacts the age distributions and the locations of the surviving clusters at all times during a merger. We conclude that accounting for the interplay between the formation, disruption, and orbital histories of clusters enables a more sophisticated interpretation of observed properties of cluster populations, thereby extending the role of cluster populations as tracers of galaxy evolution.