Cellular automaton models have long been used to study cellular processes, but may be challenging for incorporating synchronization, migratory, and high-density effects. In this work, we introduce an extension of the classic lattice-gas cellular automata, a framework which allows to consider changes in cell numbers, cell-cell interactions, migration, and evolution of genotypic and phenotypic heterogeneity. To demonstrate the utility of our approach, we consider a growing population of cells whose genotype-passed on at birth from the mother cell and subject to stochastic mutations-determines their individual proliferation rates. Using spatial simulations, we show that the model exhibits traveling-wave invasion patterns, where the fastest-growing cells accumulate at the leading edge, accelerating population expansion. We predict this behavior using a mathematical analysis based on a mean-field assumption.