Over the last two decades, metamaterials have emerged as a powerful platform for controlling mechanical and acoustic wave propagation. In particular, locally resonant metamaterials (LRMs) operating in the subwavelength regime have received considerable attention due to their strong attenuation capabilities at low frequencies. However, the analysis and design of such metamaterials for practical applications demand advanced, yet computationally efficient, modelling frameworks that can accommodate finite-sized metamaterial domains integrated with other, possibly already existing, components and structures. This work introduces a novel modelling framework that combines dynamic computational homogenization with an extended transfer matrix method (TMM) for the efficient analysis of wave propagation in locally resonant metamaterials (LRMs) with arbitrary microstructures. In contrast to existing approaches that typically rely on symmetry assumptions and normal wave incidence, the proposed framework accommodates general multilayered LRM configurations and supports both two- and three-dimensional wave propagation, including oblique incidence. Dynamic computational homogenization is first employed to extract the effective inertial and mechanical properties of the LRM, resulting in a macroscale enriched continuum representation with frequency-dependent characteristics. In particular, the effective dynamic impedance tensor captures wave attenuation phenomena in the vicinity of local resonance frequencies. The enriched continuum model is subsequently integrated within an extended TMM to investigate wave propagation across multilayered LRM assemblies coupled to acoustic and/or elastic surrounding media. Interface interactions are accurately resolved by numerically solving a constrained dispersion relation, avoiding restrictive analytical simplifications. The proposed framework is validated through comparisons with direct numerical simulations (DNS) across multiple representative case studies, demonstrating both high accuracy and substantial computational efficiency. This approach enables efficient analysis and control of wave impedance and transmission, offering new opportunities for the design of advanced acoustic devices such as filters and waveguides.