Understanding the origins of biodiversity has been an aspiration since the days of early naturalists. The immense complexity of ecological, evolutionary and spatial processes, however, has made this goal elusive to this day. Computer models serve progress in many scientific fields, but in the fields of macroecology and macroevolution, eco-evolutionary models are comparatively less developed. We present a general, spatially-explicit, eco-evolutionary engine with a modular implementation that enables the modelling of multiple macroecological and macroevolutionary processes and feedbacks across representative spatio-temporally dynamic landscapes. Modelled processes can include environmental filtering, biotic interactions, dispersal, speciation and evolution of ecological traits. Commonly observed biodiversity patterns, such as α, β and γ diversity, species ranges, ecological traits and phylogenies, emerge as simulations proceed. As a case study, we examined alternative hypotheses expected to have shaped the latitudinal diversity gradient (LDG) during the Earth's Cenozoic era. We found that a carrying capacity linked with energy was the only model variant that could simultaneously produce a realistic LDG, species range size frequencies, and phylogenetic tree balance. The model engine is open source and available as an R-package, enabling future exploration of various landscapes and biological processes, while outputs can be linked with a variety of empirical biodiversity patterns. This work represents a step towards a numeric and mechanistic understanding of the physical and biological processes that shape Earth's biodiversity.