I know that meteors that We see as shooting stars travel tens of thousands of kilometers per hour and are tiny as a grain of sand, but after hitting the atmosphere and burn I suppose that its ashes would touch the ground at some point. How long does it take? Can it be measured or observed?
I think the best model we could use for this is nuclear test fallout, whose dispersion was monitored quite extensively. It's very, very dependent on weather patterns at altitude.
For example, the Castle Bravo test mushroom cloud reached at altitude of about 40km and hence hit the Jet Stream. In addition to this the wind speeds at lower altitudes were higher than ground speeds and this was not checked for, so the combination of factors spread the fallout downwind about 280 miles (about 430 km) in about 16 hrs. This was one reason for the irradiation of many people.
What these tests and other nuclear incidents show us is that material ejected into the high atmosphere can remain there a long time, even circling the globe like traces from Fukushima.
In theory trace particles from a meteor could remain in the upper atmosphere for days, months, even years. Most would be expected to fall within days, however.