Tornadoes and dust devils display localised vortex flow.  Their fluid motion is similar to tropical cyclones but on a much smaller scale so that the Coriolis effect is not significant.  The primary flow is circular around the vertical axis of the tornado or dust devil.  As with all vortex flow, the speed of the flow is fastest at the core of the vortex. In accordance with Bernoullis principle where the wind speed is fastest the air pressure is lowest; and where the wind speed is slowest the air pressure is highest.  Consequently, near the center of the tornado or dust devil the air pressure is low. There is a pressure gradient toward the center of the vortex.  This gradient, coupled with the slower speed of the air near the earths surface, causes a secondary flow toward the center of the tornado or dust devil, rather than in a purely circular pattern.
The slower speed of the air at the surface prevents the air pressure from falling as low as would normally be expected from the air pressure at greater heights.  This is compatible with Bernoulli's principle.  The secondary flow is toward the center of the tornado or dust devil, and is then drawn upward by the significantly lower pressure several thousands of feet above the surface in the case of a tornado, or several hundred feet in the case of a dust devil.  Tornadoes can be very destructive and the secondary flow can cause debris to be swept into a central location and carried to low altitudes.
Dust devils can be seen by the dust stirred up at ground level, swept up by the secondary flow and concentrated in a central location.  The accumulation of dust then accompanies the secondary flow upward into the region of intense low pressure that exists outside the influence of the ground.