Although topological mechanical metamaterials have been extensively studied from a theoretical perspective, their experimental characterization has been lagging. To address this shortcoming, we present a systematic, laser-assisted experimental characterization of topological kagome lattices, aimed at elucidating their in-plane phononic and topological characteristics. We specifically explore the continuum elasticity limit, which is established when the ideal hinges that appear in the theoretical models are replaced by ligaments capable of supporting bending deformation, as observed for instance in realistic physical lattices. We reveal how the zero-energy floppy edge modes predicted for ideal configurations morph into finite-frequency phonon modes that localize at the edges. By probing the lattices with carefully designed excitation signals, we are able to extract and characterize all the features of a complex, low-frequency acoustic regime in which bulk modes and topological edge modes overlap and entangle in response. The experiments provide unequivocal evidence of the existence of strong asymmetric wave transport regimes at finite frequencies.