Yes, F-16s have limits in their flight manuals that cover flight into ash plumes. (different for the F100 and F110 but both are covered) It is in the 'adverse operation' section of the manual, and simply states in the first paragraph to avoid ash plumes to prevent damage, but 'if you have too...' and details instructions specific for each engine type.
Crash has it right, and Outlaw's comments are valid as well.
The biggest thing is that the melting temperatures of the fine 'glass like' ash is below the operating temperatures of most modern gas turbine engines. (F100/F110, commercial, what have you...)
The ash enters the engine, melts inside the combustion chamber, then enters the turbine in a molten state. The blades/vanes of the turbine are much cooler than the air, (otherwise they would melt too) so the ash sticks to their cooled surfaces. By reducing the vane flow, you're creating back-pressure on the combustion and compressor, which can induce compressor stalls. With this you also have an increasing operating temperature (that the FADEC tries to deal with) due to clogged vane cooling passages and choked flow. Downstream damage to the rotating hardware can also occur if the weight of the material exceeds the strength of the blades, or 'breaks off' and creates FOD within the turbine.
The ash also contaminates the small sensor openings that feed the control system, as well as get into the fuel/oil systems that will foul filters.
You have to remember the amount of air (and ash) an F-16 would swallow in only a few seconds. The buildup on the turbine could start almost immediately and cause the engine to compressor stall/flame out in very short order depending on the ash density. Jet engines are very close tolerance machines, just a few thousands of an inch in restrictions or clearances can cause massive flow distortions and interferences within the engine.
Sand can have the same effects depending on type. There are places in the world that have sand that act similar to volcanic ash. Most sand does not have the same melting effects as ash, and will not affect an engine so rapidly or badly.
And, as OL stated, 21% O2 is needed. After all the government calls them 'engine, gas-turbine, internal combustion, aviation' (Key word COMBUSTION) Without the proper fuel/air ratio the flame in the engine will stop. Engine control systems are based on the 78/21% spit of air. All work on density which is temperature versus pressure. What the FADEC, (or hydro-mechanical fuel control) doesn't realize, as the mixture it's measuring may only have 17% O2. So the 'density' of the air may be the same, but if you fly into a ash-cloud, it may contain more of the 'trace gasses' like methane, or the like. The engine still sees x pounds of air, not knowing it is 4% short on oxygen, and flame out; Or the sudden change in density can cause a stall/flame out. (Like flying through hot exhaust at 32K feet) Reducing altitude into cleaner air, or more dense air will allow the engine(s) to start again. Hopefully...
Read Boeing's comments here:
http://www.boeing.com/commercial/aeroma ... story.html
Was that what you were looking for Falconrep?
Keep 'em flyin' (in 'safe' air...)
TEG
[Airplanes are] near perfect, all they lack is the ability to forgive.
— Richard Collins