We present a comprehensive mesoscale field theory that unifies the modeling of growth, elasticity, and dislocations in quasicrystals by building on the density-wave representation of their atomic structure (De Donno et al., Phys. Rev. Res. 6(4), 043285 (2024)). Our approach is based on the amplitude expansion of the phase-field crystal model, describing slowly-varying complex amplitudes corresponding to the characteristic Fourier modes of the microscopic density field of the quasicrystal. To study the evolution of these amplitudes, we introduce a free energy functional and assume non-conserved dissipative dynamics. This approach enables the self-consistent emergence of elasticity, including both phononic and phasonic deformations, as well as defect nucleation and motion. Our theory provides critical insights into the formation of semi-coherent interfaces between misoriented quasicrystals, and offers detailed predictions on dislocation kinematics. By deriving the equations of motion for the amplitudes, we demonstrate that the elasticity and the dislocation behavior align with the predictions of classical continuum mechanics, thereby bridging the gap between the micro- and macroscopic scales. Furthermore, our theory can be regarded as a mesoscale Landau theory for phase transitions in quasicrystals, encompassing mechanical aspects. Through this framework, we establish a self-consistent connection between the microscopic quasicrystalline order and the resulting macroscopic properties, paving the way for broader mesoscale investigations into quasicrystalline systems. This work advances the understanding of quasicrystal mechanics, and sets the stage for future studies on complex mesoscale material behavior.