Cancer evolution and progression are shaped by Darwinian selection and cell-to-cell interactions. Evolutionary game theory incorporates both of these principles, and has been recently as a framework to describe tumor cell population dynamics. A cornerstone of evolutionary dynamics is the replicator equation, which describes changes in the relative abundance of different cell types, and is able to predict evolutionary equilibria. Typically, the replicator equation focuses on differences in relative fitness. We here show that this framework might not be sufficient under all circumstances, as it neglects important aspects of population growth. Standard replicator dynamics might miss critical differences in the time it takes to reach an equilibrium, as this time also depends on cellular birth and death rates in growing but bounded populations. As the system reaches a stable manifold, the time to reach equilibrium depends on cellular death and birth rates. These rates shape evolutionary timescales, in particular in competitive co-evolutionary dynamics of growth factor producers and free-riders. Replicator dynamics might be an appropriate framework only when birth and death rates are of comparable magnitude. Otherwise, population growth effects cannot be neglected when predicting the time to reach an equilibrium, and cellular events have to be accounted for explicitly.