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Or do they extend down thousands of miles, perhaps to Earth's core-mantle boundary?
Also, while scientists general agree that hot spots are fixed in position relative to the faster moving overriding plates, some recent studies have shown that hot spots can migrate slowly over geologic time.
Over a span of about 70 million years, the combined processes of magma formation, eruption, and continuous movement of the Pacific Plate over the stationary hot spot have left the trail of volcanoes across the ocean floor that we now call the Hawaiian-Emperor Chain.
Thereafter, northwesterly plate movement prevailed, resulting in the formation of the Hawaiian Ridge "downstream" from the hotspot.
The size of the Hawaiian hot spot is not well known, but it presumably is large enough to encompass and feed the currently active volcanoes of Mauna Loa, Kïlauea, Lö'ihi and, possibly, also Hualälai and Haleakalä.
Some scientists have estimated the Hawaiian hot spot to be about 200 miles across, with much narrower vertical passageways that feed magma to the individual volcanoes.
Hawaiian Hot Spot: A cutaway view along the Hawaiian island chain showing the inferred mantle plume that has fed the Hawaiian hot spot on the overriding Pacific Plate.
The geologic ages of the oldest volcano on each island (Ma = millions of years ago) are progressively older to the northwest, consistent with the hot spot model for the origin of the Hawaiian Ridge-Emperor Seamount Chain. Robinson, USGS, in "This Dynamic Planet" map of Simkin and others, 2006.