How long does it take for a stone to alter?
Mountains and rock do decompose or weather into sediment.  A basic rock cycle overview shows the possible pathways between all three rock types (igneous, metamorphic, and  sedimentary) how one type can be transformed into another. After a rock is weathered into sediment, that sediment can become a rock again.

Weathering rates of rock vary widely regional based on the following factors:


Mineral composition of the rock  (some minerals are less stable at surface conditions than others)
Amount of plant life growing on the rocks and mountains. (Plant roots can be powerful chemical and physical weathering agent)
Amount of precipitation (Chemical dissolution due to rainfall is a significant weathering factor for some rock types (carbonate rocks)
Seasonal freezing periods.  (Ice expansion in cracks and fractures can weaken rocks over very short periods of time.
Temperature range of the region.  (I believe diurnal temperature differentials specially in high mountain deserts has a specific weather impact on some rock types).
Local earthquakes and volcanic activity.
Erosion due to water (streams, rivers, lakes, and oceans).
Erosion due to glaciers and ice flows.


You can get an estimate of weathering rates of different rock types for a local region by careful examination of cemetery gravestones.