When a parcel of dry air with a given initial temperature rises, it cools at a specific rate - the dry adiabatic lapse rate. If the surrounding environment cools at a slower rate with height than the parcel, the parcel will eventually be colder than the environment and sink. On the other hand, if the environment cools faster than the parcel does, the parcel remains warmer than the environment and rises freely.
So for a parcel of dry air to continue to rise, the environmental lapse rate needs to be greater than the dry adiabatic lapse rate. Surface heating does a really efficient job of increasing low-level environmental lapse rates, which is why dust devils are more common when there's more sun.
I'm not a scientist, so I can't tell you an exact number, but the dry adiabatic lapse rate is 9.8 degrees Celsius per kilometer, or 3.3 degrees Fahrenheit per 1,000 ft. So if the environment cools at over 1 degree C per 100 meters, it will likely be conducive to dust devils.