How do mantle plumes travel from the core of Earth to the crust? As mantle plumes begin in the core and move towards the crust, I would have thought that the heat in the mantle plume would disperse to the surrounding mantle, and the plume would cease to exist by the time it reaches the crust, a large distance away.

So how is the mantle plume able to survive with such a localized trajectory on its way from the core of Earth to the crust?
The high values obtained when estimating the Rayleigh number for the mantle indicates that convection there is vigorous, and that convection is responsible for almost all the heat transported from the deep interior to the surface. Similar systems set up in the laboratory can develop 'plume-like' patterns of convective flow, and so mantle plumes are a physically plausible model of mantle convection too.