Using data obtained by the high temporal and spatial resolution Rapid Oscillations in the Solar Atmosphere instrument on the Dunn Solar Telescope, we investigate at an unprecedented level of detail transverse oscillations in chromospheric fine structures near the solar disk center. The oscillations are interpreted in terms of propagating and standing magnetohydrodynamic kink waves. Wave characteristics including the maximum transverse velocity amplitude and the phase speed are measured as a function of distance along the structure's length. Solar magnetoseismology is applied to these measured parameters to obtain diagnostic information on key plasma parameters (e.g., magnetic field, density, temperature, flow speed) of these localized waveguides. The magnetic field strength of the mottle along the ~2 Mm length is found to decrease by a factor of 12, while the local plasma density scale height is ~280 ± 80 km.