Surely most of that dust/crap is not 'stuck' onto anything, so just needs sucking out of there.
If you remove the bulbs from the lamp units this will allow access into the inside of the lamp units from the back.
All you need to do is duck tape a piece of thinnish tube to the end of the crevice tool/pipe of your hoover and shove it in there and suck all the dust out. The pipe will need to be small enough to fit through the bulb hole, maybe an old (clean) piece of hose or similar.
Worth a go before you take the lamps apart.
Mind you, the flies and sh*t shouldn't be able to get in there, so maybe a hole somewhere?