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Direct measurement of local phonon dispersion in individual nanostructures can greatly advance our understanding of their electrical, thermal, and mechanical properties. However, such experimental measurements require extremely high detection sensitivity and combined spatial, energy and momentum resolutions, thus has been elusive. Here, we develop a four-dimensional electron energy loss spectroscopy (4D-EELS) technique based a monochromated scanning transmission electron microscope (STEM), and present the position-dependent phonon dispersion measurement in individual boron nitride nanotubes (BNNTs). Our measurement shows that the unfolded phonon dispersion of multi-walled BNNTs is close to hexagonal-boron nitride (h-BN) crystals, suggesting that interlayer coupling and curved geometry have no substantial impacts on phonon dispersion. We also find that the acoustic phonons are extremely sensitive to momentum-dependent defect scattering, while optical phonons are much less susceptible. This work not only provides useful insights into vibrational properties of BNNTs, but also demonstrates huge prospects of the developed 4D-EELS technique in nanoscale phonon dispersion measurements.