In order to decide which satellite to use to trace dust deposition we need to understand the characteristics of the dust (speed of deposition, quantity of deposition in a period of time, cause of this dust, area size covered by this dust, surrounding environment, type of parent material...etc).
The dust can be cause by quarries and mining activities.
high spectral resolution radar can be used to measure the thickness of this dust every specific period of time knowing the height of the mountain before dust deposition.
To trace the dust we need know the extent of the dust in a multi-temporal multi-spectral images which can be used to classify the images considering that the area is large enough and knowing the spectral signature of this dust..
As Mr. Awad already pointed out, the problem area of tracing back the source of dust deposition is a complex problem area. In the Alps in Europe, a mountain area going up to maximally about 4800 m a.s.l. dust deposition is measured primarily by taking snow samples. Regularly long distance transport of Sahara desert fine dust occurs and part of this is deposited in the Alps. I guess that in Asia a similar phenomenon occurs in the high mountain areas of the Himalaya and the Tien Shen mountains along the Tarim basin in the Xinjiang province.
It is another business to try to track the travel path of the dust, to enable to trace back its origin. I assume that only the very fine fractions of desert sand (PM2.5 and especially PM0.1) is lifted up high enough to be deposited in the higher mountain regions of Asia which are much higher than the Alps, typically between 6000 and 8000 m. Typical sources of dust deposited in high mountain ranges are typically the very fine fraction of desert sands, loess dust bowls and volcanic fine dust which is on eruption sometimes injected straight in the stratosphere enabling it travel worldwide. The residence time of dust in the stratosphere can be even a few years. To trace back the origin of this type of dust is not an easy task for sure.
On the other hand the PM10 fraction will probably never reach higher mountain ranges, because it will have gravitated before ever reaching the permanent snow ranges of the Himalaya, Tien Shen or Alps for that matter.
So I guess I guess that deserts and volcanoes are the main sources of the dust you will find in high mountain ranges. I think this narrows down a little bit the sources you have to look for. To complicate this matter even more the emission form volcanoes or deserts is very variable in time hence episodic. that does not make it easier to trace these dust tracks down using remote sensing. Mr. Awad gave some suggestions of RS techniques which can be applied. Add to this AOD determinations using hyperspectral RS above very dark surfaces or in the blue range of the EM spectrum, and you might be able to pick up the traces of stratospheric dust tracks.