In this paper, we explore how potential biomechanical influences on cell cycle entrance and cell migration affect the growth dynamics of cell populations. We consider cell populations growing in free, granular and tissuelike environments using a mathematical single-cell-based model. In a free environment we study the effect of pushing movements triggered by proliferation versus active pulling movements of cells stretching cell-cell contacts on the multi-cellular kinetics and the cell population morphotype. By growing cell clones embedded in agarose gel or cells of another type, one can mimic aspects of embedding tissues. We perform simulation studies of cell clones expanding in an environment of granular objects and of chemically inert cells. In certain parameter ranges, we find the formation of invasive fingers reminiscent of viscous fingering. Since the simulation studies are highly computation-time consuming, we mainly study one-cell-thick monolayers and show that for selected parameter settings the results also hold for multi-cellular spheroids. Finally, we compare our model to the experimentally observed growth dynamics of multi-cellular spheroids in agarose gel.