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We use the Dynamic Density-Functional Formalism and the Fundamental Measure Theory as applied to a fluid of parallel hard squares to study the dynamics of heterogeneous growth of non-uniform phases with columnar and crystalline symmetries. The hard squares are (i) confined between soft repulsive walls with a square symmetry, or (ii) exposed to external potentials that mimic the presence of obstacles with circular, square, rectangular or triangular symmetries. For the first case the final equilibrium profile of a well commensurated cavity consists of a crystal phase with highly localized particles in concentric square layers at the nodes of a slightly deformed square lattice. We characterize the growth dynamics of the crystal phase by quantifying the interlayer and intralayer fluxes and the non-monotonicity of the former, the saturation time, and other dynamical quantities. The interlayer fluxes are much more monotonic in time, and dominant for poorly commensurated cavities, while the opposite is true for well commensurated cells: although smaller, the time evolution of interlayer fluxes is much more complex, presenting strongly damped oscillations which dramatically increase the saturation time. We also study how the geometry of the obstacle affects the symmetry of the final equilibrium non-uniform phase (columnar vs. crystal). For obstacles with fourfold symmetry, (circular and square) the crystal is more stable, while the columnar phase is stabilized for obstacles without this symmetry (rectangular or triangular). We find that, in general, density waves of columnar symmetry grow from the obstacle. However, additional particle localization along the wavefronts gives rise to a crystalline structure which is conserved for circular and square obstacles, but destroyed for the other two obstacles where columnar symmetry is restored.